Cane Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⡶⠶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⣻⡏⣰⣾⢈⣈⣉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣥⡠⣿⠟⠛⠛⠻⡟⠋⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠰⣿⡟⠁⠀⠀⠀⠀⣿⡆⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⢱⣴⡟⠀⠀⠀⠀⠀⢠⡏⠙⣾⣿⠀⢀⣤⠶⢫⢿⠿⠓⢲⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠹⡿⠁⠀⠀⠀⠀⠀⠀⠑⠒⠾⢋⣴⣿⡋⢀⣶⣏⣤⣿⠈⡉⠻⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣤⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⣾⣿⠞⠋⠉⠉⠙⢿⠿⢳⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢻⣼⣴⡟⠁⠀⠀⠀⠀⠀⣿⡦⠈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣷⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⢈⡿⠋⠀⠀⠀⠀⠀⠀⣼⠋⢣⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⢹⡿⠀⠀⠀⠀⠀⠀⠀⢀⣼⢙⣷⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠑⠦⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⣄⣾⠁⠀⠀⠀⠀⠀⠀⣠⡾⣧⣨⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⢸⡇⠀⠀⠀⠀⠀⢀⣴⠟⠹⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢯⣿⣿⠀⠀⠀⠀⠀⢠⡾⢟⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣄⣹⠇⠀⠀⠀⠀⣰⡿⡃⣀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⣷⣿⡟⠀⠀⠀⢀⣾⡟⢷⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠈⣿⠁⠀⠀⢠⣿⣏⣧⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠻⣳⣿⠃⠀⠀⣴⡏⠘⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⣼⠏⠀⢀⡾⣿⢦⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⣿⣇⣨⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠛⣹⣧⣬⡾⠋⣹⣄⠀⠀⠀⠀⠀⠀⣰⡟⢙⣏⣠⣾⣉⣉⣻⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠉⢿⡿⠋⠉⠹⣿⠛⠻⣆⠀⠀⠀⠀⣼⢿⣧⣸⠟⠉⠉⢻⣏⠙⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣳⣼⡇⠀⠀⠀⠘⣿⣶⣿⡆⠀⠀⣰⣇⢘⡿⠁⠀⠀⠀⢸⡿⣄⣾
⠀⠀⠀⠀⠀⠀⠀⠻⠬⠛⠃⠀⠀⠀⠀⢹⣦⣤⣽⡄⢠⡏⢻⣿⠃⠀⠀⠀⠀⠘⠤⣘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠐⠦⣀⣻⡉⠛⣻⣾⢡⣴⠇⠀⠄⠊⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⠀⣌⣉⣙⣿⢌⣟⡋⢁⡤⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⢈⣀⣠⣴⠶⢿⣿⣿⡷⠶⠦⣤⣠⣀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠑⠁⢠⣾⠃⣽⠷⡄⠀⠀⠈⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⡀⣼⠉⢺⣿⣤⣿⠀⢶⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢮⡞⢰⣿⣶⡿⣧⣈⢛⣇⠘⣿⢋⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢇⣹⠇⠹⡷⠿⣿⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠻⣿⡟⠀⠀⢻⣄⢨⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣀⣽⠁⠀⠀⠈⣿⡛⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢉⣿⠃⠀⠀⠀⠀⢸⣷⣲⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠀⠙⠒⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢃⠀⠀⠀⢠⠔⠁⢀⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣄⠸⡇⣾⢀⢄⡴⠖⠁⢠⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠉⠀⠈⠉⢿⣆⣿⣧⢏⠈⠀⣀⠤⡟⠒⠒⣂⠚⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⢀⣄⡸⢷⣏⡇⠀⠄⠁⠰⡇⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡋⠅⠲⢟⢻⣿⡗⢀⡨⠄⢦⣷⢇⡾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⢀⣴⠷⣼⡟⠀⠀⠀⢸⡏⡌⢀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⡐⠏⠁⠀⠟⣃⣀⣄⢀⠈⢡⡴⢃⡤⠂⠀⠁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⢀⠴⣾⠇⠀⠀⢨⣀⡾⠳⣋⠤⠤⠄⠀⠀⠤⣀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⣏⠀⢀⡤⢺⣿⡿⠟⠶⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣿⢢⠋⠀⣸⠟⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡁⠀⠀⡇⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⠇⠀⢸⡿⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢸⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢛⢸⣧⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⣸⠀⢸⠋⠀⢀⡤⠒⡞⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢸⠀⢸⠀⣠⠋⠀⢘⣀⡤⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⢸⡘⡿⢠⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠒⠲⢤⣿⠘⢸⡷⣷⠁⣼⠀⠀⠀⣤⣠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠤⠤⣿⠀⢸⣇⣇⣀⣿⠤⠶⠟⢫⠀⠀⠀⢀⡀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⣀⣈⣭⢌⣽⣨⡤⠤⠤⠶⠖⠚⢻⡟⠿⠋⣀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⢀⣀⡀⠴⠛⠉⠰⠉⠖⠀⢀⠀⠘⣿⢶⠲⠶⠶⠤⠿⠧⠤⠄⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠁⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠈⠉⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠆⠀⠑⠀⠀⢰⠀⢠⠁⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⠀⠀⠁⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠈⡃⠀⠀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⢀⠀⢀⣿⣷⠀⠀⠀⡠⠀⠀⢹⠀⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠠⠤⠤⣿⣆⠀⠀⢳⠀⢿⡇⣶⡄⠘⠀⠀⠀⢸⡀⠀⠰⠀⡸⠃⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊⠀⠀⠄⣀⡀⠔⠚⠉⢿⡟⢦⡜⢳⣼⠁⣿⠗⡂⠉⠀⠀⢹⠟⠲⣣⣐⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠉⠉⢂⡀⠀⠘⣯⠀⢁⠈⠇⣴⡇⢻⡏⢦⠀⠀⢴⠈⣫⠏⠙⠷⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠳⢤⣀⠀⠑⠀⣀⣿⡀⠀⠀⣄⣿⠁⣨⠏⠠⡎⠀⣸⣟⠁⠀⠀⠀⠀⠄⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⣸⣅⠸⣧⣤⠀⠹⡧⠀⠀⣶⠀⢸⠀⡿⣡⠔⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠘⡏⡈⠀⢹⠉⣧⠀⠁⠰⠀⢿⢀⡌⢸⣿⠏⠀⠀⠀⠙⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣈⡟⢻⣆⣾⠀⣿⠀⠃⢂⣠⢀⣼⢇⣿⣤⣤⡀⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠤⠀⠉⠋⠉⠋⠻⢶⣿⣷⣞⣠⡄⠘⠛⠃⠺⢩⠚⢁⠈⠀⣿⠓⠲⠄⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠀⠁⠀⠀⠀⠀⢠⡞⠙⢿⡿⠛⠁⡀⠐⠁⣀⠏⠀⢸⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⡀⠀⠀⠀⠀⠀⠀⢀⡌⣹⠁⣐⢸⡇⠀⢠⠁⠀⠀⣹⡀⠴⠾⠶⠶⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⣀⣄⠹⣧⡇⣼⠃⢸⡶⠃⣸⠀⠀⢰⠏⠀⠀⠆⠀⠀⢸⡏⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⠛⢦⣻⣷⡀⣤⣏⠀⠀⢹⠀⠰⡿⢻⡆⠀⠀⠀⠀⠀⠇⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢧⠈⠁⠉⠀⠀⠹⣿⣥⠉⣿⠀⠀⠾⠀⣸⠃⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠉⡇⠀⣿⠀⢀⡄⠀⣿⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⢻⠀⢿⡀⠀⡅⠘⡿⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⡆⢸⡇⠐⡇⢸⡇⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠘⣷⠰⡇⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠸⣧⣿⠀⣶⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣥⣿⣀⡿⣿⣿⣎⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣼⣿⣿⡂⣼⣿⣿⡥⠤⡀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡧⢿⣿⡟⠻⢷⣿⠿⠿⢿⣷⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠛⠒⠓⠛⠉⠉⠁


⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣶⣶⣶⣶⣾⣶⣶⣶⣾⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣾⣿⣿⣷⣶⣿⣿⣷⣶⣷⣶⣶⣶⣿⣶⣾⣶⣶⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣴⣇⣠⣾⣦⡀⠀⠀⠀⠀⠾⡷⢤⣷⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠋⢹⡿⠉⡙⣿⣿⣿⣿⣿⣫⣈⣿⢉⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⡈⠋⠀⢰⣄⡇⠀⠀⠀⢰⡾⠀⢠⣤⠇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢷⣾⣿⡟⠻⣿⣿⣿⣿⡟⢉⣿⣿⠟⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠛⠋⠀⠈⡛⢷⠀⠀⠀⣩⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣤⣽⣿⣿⣤⡸⣿⣿⣿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣬⠀⠀⠘⣋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⠛⣿⣿⣯⡴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡙⡇⠀⡾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣦⣿⣿⣃⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡇⢠⣶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣈⢹⣿⢋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣹⠈⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣮⣭⣭⣯⣽⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣾⣶⣿⣾⣦⣄⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠢⡀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠈⠠⠀⠀⡄⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣓⠀⠒⠂⠠⢄⡀⠋⡀⠀⠄⢰⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠤⣬⣷⣾⣤⣧⢸⠀⠘⠀⢀⣠⣶⡟⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣤⠀⡰⠋⠀⣿⠁⠀⢀⠔⠋⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠈⠻⣿⡇⢰⡗⠶⠊⠉⢣⠀⠀⠲⡖⢀⠀⠄⠀⣰⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠉⠉⠛⠛⠛⠶⢦⣄⠀⠀⠘⡇⢸⣾⡀⠀⠀⠐⣤⣴⣶⣿⠄⠒⠢⠞⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠶⠾⠶⠟⠻⠷⣤⡠⢸⠛⠉⠑⡀⢴⣿⡿⠋⠁⢃⠀⠀⠲⠀⡀⠠⠄⠠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣯⠀⢮⡩⡧⣾⣿⣄⣀⣀⢸⣶⣶⣷⡤⠔⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⣿⠀⠀⠏⢀⣄⢼⣄⠸⢿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠚⠈⣶⣤⣾⣏⡥⠒⣲⡇⡟⠁⢀⣤⠴⠒⠛⠛⠲⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⢿⠓⢀⣠⣿⣿⣿⣿⢀⡤⠉⢣⡴⠚⠉⠉⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⡀⡀⠈⠈⢻⣭⡿⠿⣯⣼⠚⠛⣉⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡄⣀⠙⢻⣷⠚⠻⠙⣿⠉⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⣦⣄⡈⢣⣧⣠⢸⡇⠀⢰⠀⠻⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢸⡇⢹⠀⢸⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡇⢸⣄⣰⡀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢹⡟⢻⠃⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⣄⠘⡇⠀⠇⢸⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣏⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢸⠀⢇⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⢸⣆⣹⣼⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡄⢻⡟⠿⢿⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⡗⢸⡇⠰⡞⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⢸⡇⢰⣇
⠀⠀⠀⠀⡔⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠁⢸⡇⠀⡄
⠀⠀⠀⠘⣧⠈⢁⡀⢀⠀⣀⣀⠀⡀⣀⠠⣀⡀⠀⢸⠃⠀⢀⣀⡀⢀⠀⡀⢀⣄⠀⠀⠀⢸⠀⠀⢸⣷⣾⣷⠀⡇
⠀⠀⠀⠀⡌⢳⢸⡇⣿⢠⡇⣽⢸⠁⣿⠀⢸⡇⠀⢸⡀⢀⣸⢀⡇⢸⠀⡇⢸⠜⠀⠀⠀⢸⣧⣼⢹⡿⠻⡟⠻⡇
⠀⠀⠀⠀⠁⠈⠈⠃⠉⠈⠁⣿⠈⠁⠉⠀⠈⠁⠀⠀⠁⠉⠈⠀⠁⠉⠀⠁⠈⠀⠀⠀⠀⠈⡇⢹⢸⡇⢠⠇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣢⣇⣸⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⡇⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⠇⠀⠀⠁⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⢄⠈⣀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠀⠀⠀⠰⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡿⣀⠶⠀⠀⠈⢻⡄⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⠺⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢡⡄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠘⠉⠻⣄⠀⠄⠀⠹⣆⠀⢀⠖⠀⡀⠀⠀⠀⠿⠁⠀⠀⠀⠀⢤⠐⠂⠉⠉⢉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⠈⠁⠈⠳⣄⠀⠂⢹⡆⠈⠁⠀⠀⠀⠌⠀⠀⠀⠀⢀⠄⠈⠀⠀⠀⠀⠀⢸⡇⠠⠄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠈⠑⠄⠘⡧⡀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠋⣉⣁⣈⠀⠀⠠⠀⠀⠀⠀⠱⣤⡀⠀⠀⡈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠠
⠀⠀⠀⠀⠀⠀⠀⠔⣁⠔⠋⠉⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⢰⣭⡁⠀⠀⠀⣉⠞⠁⠠⠴⠶⠆⢄⠀⠀⠀⣘⠇⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠑⢆⠀⠀⣿⠅⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠲⣇
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡆⠀⠀⠀⠀⡞⠃⠀⢈⠀⡄⢠⠏⠀⠀⠀⠀⠔⠒⣦⣄⠀⠀⠰⣰⠀⢀⠄⢀⠔⠀⠀⠀⠋
⠀⠀⡀⠁⠀⠀⢤⡀⠀⠀⠃⠀⠀⠀⠀⠇⠀⠀⠀⠐⠀⠀⠀⠛⠃⠁⠘⠀⠀⠀⡀⠀⠀⠛⠋⠈⢄⠀⣷⠃⣴⠏⠀⠀⠀⠀⠀⠀⡀⠾⠃
⢀⠜⠁⠀⠀⠀⠀⠉⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠀⠈⠁⡸⠐⠋⠀⠀⠀⠀⠀⠀⠢⠊
⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡑⢄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⡰⠁⠀⠀⢴⠆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠁⠀⠣⡀⠀⠀⠀⠘⣘⠀⠀⠀⢀⠀⢿⡄⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⣸⡇⣼⠇⠈⣠⡴⠁⣌⢁⡄⢀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⡀⡏⠀⠀⠔⠀⠀⢈⠁⠀⠀⡤⠀⠀⠐⠁⠔⢶⣿⢠⡟⠀⠉⣠⠊⠈⢀⡄⠉⣡⡄⠉⠙⡏
⠀⠀⠀⠘⠛⠛⠛⠒⠂⡀⠀⠀⠀⢰⡌⠇⠀⢀⡇⠀⠊⡀⠁⠀⢨⣦⡐⠎⠀⣠⣤⠀⠊⠀⠈⢿⠘⠃⡄⠸⠃⡴⠃⢀⠄⢚⠁⡇⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠹⡄⠀⣿⠀⠀⠰⠀⠀⡄⠰⠛⠏⠀⠈⠹⢦⡉⡶⠈⣁⣽⣅⣆⣰⡇⠈⡀⠞⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢂⠐⣧⢸⡿⡀⢀⠇⠀⠀⠀⠀⣶⠀⠀⢀⡠⠀⠹⢁⡆⠙⠉⣻⣿⠏⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠴⣄⡀⠀⠀⣸⣧⠀⢨⠀⡀⠂⠀⠠⠆⠀⠰⠋⠀⠐⠀⠺⠀⠀⠀⡿⠃⡜⠁⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⢀⣤⣤⣄⣤⣄⡀⠚⢧⡈⢢⡆⠙⣿⢀⡖⠰⠀⠀⠠⠀⠸⠶⠀⠀⢀⠀⠀⠀⠀⠸⡆⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⡘⠋⠁⠀⠘⢉⠒⢩⠒⠄⢱⡄⢳⡀⠁⠘⠀⠀⠀⠂⠀⠀⢀⣈⠀⠀⠀⠀⠀⠀⠀⠰⣥⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⣦⠁⠀⠀⠐⠀⠂⠀⠀⠑⢄⠉⢿⡀⢿⣦⡀⠀⠠⡾⠋⠙⠃⠤⠉⡀⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⢻⠈⣿⠀⠀⠐⠀⠀⠀⠀⣴⢾⠄⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢰⢢⡔⠀⠀⠀⠀⠀⠀⠀⣄⠃⠀⠀⠀⠀⠀⠀⢠⡄⠸⡆⠀⠀⣿⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠛⠓⠸⠦⠀⠀⠀⠀⠘⠁⠀⠃⠀⠀⣤⠀⢸⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠛⠀⠀⠘⠃⠀⣿⠀⡈⠀⠀⢀⠋
⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢰⡆⠀⢀⡈⢠⢀⣿⠀⠁⠀⠀⡈
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⣀⠀⠀⠈⠃⢸⣼⠏⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢶⡾⠃⠀⠀⠀⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢰⡦⠀⠀⡇⢾⠃⡄⠀⢀⡠⠀⠐⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⢶⠦⠀⠀⠀⠀⠀⠀⠀⠀⣇⠠⠘⠀⢢⠊⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⢰⡟⢲⢁⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠃⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⢠⣤⠀⠀⠆⠁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⠀⠀⠀⠀⢠⣀⠀⠀⠀⣀⡀⢀⣾⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⠀⠀⠀⠀⠈⠉⠀⠀⠈⠛⢀⢢⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣀⠀⠀⠀⢸⡦⠀⠀⡷⠀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢂⠀⠀⢀⣀⠀⢰⣴⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣐⡎⢠⣄⣈⠹⠶⢀⣶⡴⠟⠋⠉⠛⠻⠶⢦⣤
⠀⠀⠀⠀⠀⠀⡀⠀⠴⠆⠛⠋⠀⠀⠀⠀⠚⠛⠉⠉⠙⠛⠋⢥⣤⣤⡴⠶⠶⠿⠟⠁⠈⠁⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠰⣤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢀⣀⣀⣀⣀⡈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠛⠈⠉⠉⠉⠉⠉⠉⠈⠛⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⡰⢇⠀⡀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⠄⠀⠀⠀⢠⡾⠁⠈⢦⡐⢄⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⠂⠀⠀⣀⡴⠋⠀⠀⠀⠀⠙⣄⠉⠢⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⢰⡶⠤⠤⢒⡭⠚⠁⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠞⠋⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠸⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⢸⠂
⠀⠀⠀⠀⠀⠀⢀⠴⣶⠃⠀⠠⠀⠸⠿⠀⠀⣠⠀⠀⠀⠀⠈⡟⠃⠀⠀⠀⠹⡾⠦⣄
⠀⠀⠀⠀⠀⠀⢸⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠈⠓⠼⢧⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡾⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠊⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣲⠤⣄⣀⣀⣤⣤⣄⣀⠠⡴⢾⠉⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⢻⣍⠀⠀⠀⠀⠀⢀⣴⡏⠀⠀⠀⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⡇⠀⠀⠀⠸⠇⠉⠒⠒⠒⠊⠁⢸⡇⠀⠀⠀⠀⠀⢠⠟⢢⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢶⡇⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⢀⠟⠀⠀⠹⣆
⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠘⡷⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢡⢸⠁⠀⠀⠀⠹⡆
⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠻⣀⠀⠀⠀⠀⢸
⠀⠀⠀⢠⠁⠀⠀⠀⠢⣀⣼⠀⠀⠀⠀⣀⣼⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⡆
⠀⠀⠀⢸⠀⠀⠀⠀⢸⠏⢸⠀⠀⠀⠀⢈⡍⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⢘⡦⠀⠀⠀⠀⡇
⠀⠀⠀⠈⡆⢀⣤⣐⣺⡀⢸⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠘⣄⠠⣾⠉⠉⠻⠗⡂⠀⠀⢰⠁
⠀⠀⠀⠀⠘⢿⡞⠉⠉⠑⡾⡀⠀⣀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣄⡄⢀⡾⢁⣀⣰⠃
⠀⠀⠀⠀⠀⠈⠳⢤⣤⡞⠁⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠗⠒⣸⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⠁⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⣿⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣟⠃⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⢿⣄⣴⠞⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠘⠀⠀⠀⠈⠉⢿
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⡀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⡴⠋⢀⣠⡤⠶⠚⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡿⠲⠦⠤⠤⠤⠤⠤⠴⠶⠟⠛⠂⠉⠉⠀⢀⡀⠴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢱⡤⢤⣀⣀⣀⣀⣀⣠⣴⠶⣥⠤⢴⠒⠉⠁⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⡄⡇⠀⠀⠘⡄⠀⠀⠀⠉⠈⡏⠉⠀⠀⠀⠀⠸⡀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⢻⡀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠀⠀⢏⠀⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠈⠳⣤⣤⢘⡧
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⠛⠲⢦⣄⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠀⠨⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢦⣶⣴⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢰⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠄⢀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣫⡟⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⣸⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⢾⠟⣩⣾⣿⣇⢛⠛⣛⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠰⣧⠀⠀⠀⢱⠀⢀⡀⠀⠀⠀⠀⠀⠀⠐⠊⠷⢶⣧⣠⣿⣡⠟⠻⠶⢿⣿⠷⠄⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢌⣆⠀⢄⣿⣡⡐⠁⢈⡀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠚⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⡀⢿⣷⣦
⠀⠀⠀⠀⠀⠊⠀⠐⠲⡀⠘⡟⠟⢟⡿⣧⡀⢨⣯⡷⢆⢀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⠀⣀⣀⡀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠂⠈⠉⠐⢠⡀⠀⢡⡈⢲⡇⢋⢺⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⣯⢩⢛⣄⡄⠸⣤⡾⠋⠀⠐⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⢀⠰⠀⠀⠛⢶⣿⣮⡔⠚⠛⡟⠉⢹⠒⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⣨⡇⠀⣸⠀⠀⠀⢀⣀⣠⣇
⠀⠀⠀⠀⠀⠁⠀⠀⢠⡐⡏⣼⠏⠀⠀⣠⠠⠦⠬⣇⠀⠀⠀⠀⠀⢀⣄⣀⡠⠀⠀⣼⣦⣄⡀⠀⠀⠀⠀⠤⠀⠼⡧⠰⣧⡀⠀⠀⣹⣿⣿⣿⣶⡤
⠀⠀⠀⠀⠀⠀⡔⠙⠢⣿⣟⣸⠃⠂⢠⢧⠀⠀⠀⠃⠡⠀⠀⠀⠤⡾⢉⣭⣴⣿⣿⣿⣟⡿⠗⠀⠀⠀⠀⠐⠒⠊⠀⠀⠈⠉⠉⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⠿⢹⠀⡃⣸⠈⡇⠀⠀⠀⠀⠀⠀⠀⠘⠳⠏⠿⠿⠛⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠸⡸⡄⡄⡇⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡄⣤⠤⣤⠤⠠⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡇⢲⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⡀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⣿⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣶⣾⠻⣟⣠⣾⣥⣶⣾⣷⣿⣿⣶⡖⠋⠛⠛⣓⣻⣧⣿⡧⣧⣶⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⣿⣿⣿⣿⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡷⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣷⣿⣿⣿⣷⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣸⣧⣤⣤⣤⣴⣶⣾⣟⣿⣭⣤⣤⣤⣤⣴⣿⣭⣀⣀⣀⣀⣨⣧⣤⣤⣄⣤⣧⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠉⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠐⠀⠉⠠⣾⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⢹⣷⠀⠀⢀⠀⠐⡆⠀⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠐⠲⣿⣆⡀⢰⡀⡯⢰⡄⠁⡀⢀⣇⠀⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⡲⢆⠉⠀⢻⡝⢯⢹⡇⡻⣯⣄⠀⠀⣚⢭⠳⢦⠀⡀
⠀⠀⠀⠀⠀⠀⡐⠀⠒⠲⠶⣄⡀⠙⠄⢈⣇⠘⢠⣸⡇⢸⠎⡥⠈⣿⡟⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⢦⣸⣅⢿⣦⡀⠻⠄⠡⡆⢸⢰⣏⡔⠀⢀⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⢿⣄⢸⡅⡇⠀⠀⠀⠃⣼⣾⠋⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠐⠒⠲⢻⣼⣮⣷⣉⠈⠶⠞⡟⡿⡛⠛⢷⠦⢄⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠈⠀⠀⢀⢸⡎⠻⡿⠛⢀⠀⠀⡸⠀⢀⠀⢸⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⢀⣆⡼⢠⠏⣇⡄⡆⠀⢠⠗⠊⠛⠙⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠙⠦⣹⣷⡉⣸⠋⠐⡇⠠⡾⣆⠀⠀⠀⠀⠃⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠨⠆⠈⠈⠀⠘⢿⡼⢹⠀⠰⠇⢸⠃⢻⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⡇⢸⡄⠰⠀⣾⠀⠘⢧⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⢹⠈⡇⢠⠠⡏⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣿⠐⢳⡇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣹⢿⠰⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣊⣿⣿⡃⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⢿⣿⣿⢷⢿⣿⣿⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣰⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⢰⣶⣶⣶⣶⣶⠀⠀⠀⠀⢀⣤⠄⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⣈⣁⣠⠀⢀⡀⢠⣶⣾⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠿⣿⣾⣿⡿⣛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⡀⠀⠠⢼⣎⢳⡑⣤⣤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠿⢉⣴⠋⡄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣛⡙⠿⠿⠛⠀⣤⣤⣤⣶⣿⣦⢹⣌⢿⣿⠏⠀⠀⠀⠀⠀⠀⣧⠀⢤⣾⡿⠟⣉⣴⣿⣿⠀⣷⠦⠶⡾⠃
⠀⠀⠀⠀⠀⠘⠐⠻⠿⠀⠠⢶⣶⣾⣿⠿⣟⣻⣿⣿⢧⠹⣦⠟⠁⠀⠀⠀⠀⣀⡔⠁⠀⠠⠛⠀⢸⡿⠛⢿⣿⣇⠐⢀⣀⢰⡀
⠀⠀⠀⠀⠀⠀⠠⣆⢠⡶⠺⣛⣭⣽⣿⠿⠛⠛⠋⠉⠀⠉⠁⠀⠀⠀⠀⠀⠙⠋⠻⠷⢶⣦⣄⣠⣿⣀⣰⠟⠋⠛⠿⠿⣿⣿⠷⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠔⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠳⠤⣶⣶⣶⡶⠟⠁

⠀⠀⠀⠀⠀⠀⣘⣛⣉⣉⣉⣹⣞⣛⣒⣒⣋⣉⣻⣿⣿⣯⣭⣭⣽⣿⣿⣛⣂⣛⣉⣁⣁⣿⣛⠛⠉⠉⠉⢙⣋⣉⣉⣉⣽⣉⣉⡁
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⡤⠤⠤⢤⠄⢠⠤⠤⠰⠶⠆⠂⠒⠀⠀⠐⠐⠒⠒⠒
⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠤⠠⠠⠀⠀⠄⠤⠀⠠⠤⠀⠄⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠠⠄⠀⠤⠀⠄⠀⠠⠀⠤⠀⠠
⠀⠀⠀⠀⠐⠂⠲⠒⠒⣀⢐⣒⣂⢀⠂⣒⣒⢀⣐⣒⣒⡀⣀⣒⣒⣀⠐⣒⣂⣀⣀⣀⣀⣀⢀⣀⣀⠀⠀⠀⠀⣂⠐⠀⢀⡀⣌
⠀⠀⠀⠀⢠⣭⣭⡿⣨⡟⢸⣟⣋⣡⣤⣡⣀⣸⣻⣿⡛⣧⣿⣿⣹⣧⣶⣶⡋⣄⣒⣒⢒⣂⣒⡒⣒⣀⡒⣒⠀⡄⣒⣚⢓⡑⠈
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣷⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢿⣿⣿⣿⣧⡠
⠀⠀⠀⠀⠀⠈⠻⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂⠀⠂⠀⠀⠀⠀⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⢃⣤⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠈⢀⡴⢁⠤⠂⠀⠀⠀⠀⠀⠀⠀⢀⢸⠅⠘⡻⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢀⠀⣉⣧
⠀⠀⠀⠀⠀⠀⡔⠁⣴⠋⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⣠⡈⠂⠀⠝⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⢣⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠐⣠⠞⠡⠂⠁⠀⠀⠀⠀⠀⠀⠀⢠⡞⠃⢠⠨⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠐⠢⠜⠋
⠀⠀⠀⠀⣠⣾⡫⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣞⡛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣡⠄⠊⣀⡀
⠀⠀⠀⢀⠔⠉⠀⠀⠀⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠈⠙⠘⠁⣟⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⠀⠀⢿⣷⡴
⠀⠀⠀⠈⣆⡀⠸⢰⠄⢸⠂⣃⡈⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⡀⣠⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⠿⠀⠀⣼⢧⡀
⠀⠀⠀⠀⠀⠑⠀⠍⠀⠠⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠈⠁⣟⡄⠀⠀⠀⠀⠀⠀⠰⡇⠠⠈⠓⢬⠇⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⢤⠀⠒⢀⡧⠄⠀⠀⠀⠀⠀⠀⠀⠁⢢⣠⠇⠐⣾⠿⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⣀⡠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⣀⢠⠀⠀⠀⠐⠄
⠀⠀⠀⠠⡄⠄⠽⠞⠛⠒⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⣷⢉⣹⣿⣄⣌⢠⣤⡇⠀⠀⠀⠀⠀⣴⠁⠀⠄⠂⠀⠐⠂⠀⠠
⠀⠀⠀⠀⠁⢠⡖⣤⣀⢢⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⣟⠃⠀⢽⡄⠀⠈⠁⠀⠀⠀⠀⠀⠁⢴⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠆⠈⠈⢙⣏⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠔⠐⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⢤⡄⣁⡀⠀⠂⠃⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠑⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡧⠤⢤⠝⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⣤
⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⣂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⣾⡅
⠀⠀⠈⠀⠀⠸⡇⠄⢀⠰⣷⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣆⢰⡌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⣴⠿⢿⣤
⠀⠀⠀⠀⠀⢠⡇⠀⠨⠄⢻⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⡸⠛⢠⠈⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠻⠁⠀⠨⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠠⢥⣝⣀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⢰⣶⣛⣛⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⢤⡤⠾⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠀⠄⢀
⠀⠀⠀⠀⡠⠈⠁⠁⣀⣠⡶⣆⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠐⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⡄⠀⠀⠐⢢
⠀⠀⠀⠀⣇⣈⡤⠞⠋⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⣾⣿⣦⣤⣦⣼⣤⣶⢴⡆
⠀⠀⠀⠀⣿⠃⠀⠐⠀⠠⣶⡏⠀⠀⠀⠀⠀⠀⠀⢈⠛⠀⠊⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠤⠲⣦⣀⡟⠒⠒⠿⣶⣦⠉
⠀⠀⠀⠀⠈⠀⢀⠀⠀⠄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠠⠆⠀⠐⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠉⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠶⠶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⠶⠶⢦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢹⣸⡏⢹⡛⠛⠻⡟⠶⠶⢶⠶⠶⠶⡶⠶⠶⣶⠶⠶⢲⡖⠞⠻⡟⠋⢻⢉⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣧⣇⣀⣇⠀⢘⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⣸⠀⠀⢸⡇⠀⢘⡇⠀⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣉⣙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣻⣏⠁
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠈⠉⠉⠉⠉⠙⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋⠉⠀⢻⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣸⠇⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡆
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⠟⠀⠀⠀⠀⢸⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠘⢷⣤⣄⡀⠀⠀⠀⠀⢀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⣿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠸⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢿⡍⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢹⡟⠀⢸⡏⠀⠀⠀⢸⡀⠀⠀⠈⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡴⠟⠀⠀⠸⠷⣤⣤⠶⠾⠷⠶⠶⠶⠋
⠀⠀⠀⠀⢠⡴⠶⠶⠶⣶⠶⣤⣤⣈⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣴⠶⠲⢶⣄⡀
⠀⠀⠀⠀⣿⠀⠀⠀⢰⠃⠀⠀⠀⠉⢻⡄⠀⠀⠀⠀⢀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠉⠀⠈⣿⣄⠀⣠⠞⠛⢶⣄
⠀⠀⠀⠀⠙⢷⣄⣀⣸⣄⠀⠀⠀⣀⡾⠃⢀⡀⠀⢰⡟⠁⠀⠀⠙⣳⣦⣀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠉⠳⣄⣴⠟⠙⢷⣇⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⣛⣯⡶⠛⠋⠙⢷⡈⠻⣄⠀⣠⠞⠁⠀⠙⢷⣄⠀⠀⠀⢸⡅⠀⠀⢀⣴⠟⠁⠀⠀⠀⠙⠷⣤⣀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠳⡄⠀⢀⣼⠇⠀⠈⠻⣧⡀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠙⠶⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⣀⣿⠶⠟⠁⠀⠀⠀⠀⠈⠻⣦⣀⠀⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉

⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠓⢲⣶⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣆⠀⠀⣻⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⢄⣠⠉⠉⢻⠃⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⡠⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠹⣤⣤⠆⠀⠀⢀⣶⣾⣿⠀⡆⣀⠤⡔⣁⡀⠀⢢⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡊⠀⠀⠀⠀⠈⠑⠀⠀⠀⠈⠈⠉⣷⢾⠳⠁⠁⠌⠓⠈⡶⠑⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣑⣄⣀⠀⠤⠄⠀⠀⢀⣀⣀⣼⠟⠢⡎⠳⠐⠏⢴⣾⢥⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢠⠃⠀⠀⠀⠁⠢⠀⠀⠀⠾⠿⢿⡿⣀⡦⠓⢤⠔⢋⡡⠄⠀⠫⡁
⠀⠀⠀⠀⠀⠀⠈⠑⢄⡀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠈⠽⠟⠚⠚⠿⣗⠀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⠔⠢⢶⡢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠞
⠀⠀⠀⡠⠃⠀⠀⡀⠄⠀⠁⠀⠀⠀⠀⡠⢲⠀⣬⣤⣀⣼⠈⠢⡁⠀⢠⣀⡀⠀⡀⠀⡀
⠀⢠⠊⠀⠀⢶⢋⠤⢤⠔⠁⢀⡀⣴⠊⠀⠘⣴⠿⠿⣿⠃⢸⠁⠘⢄⠔⠃⢍⠉⠁⠀⡵
⠀⠀⠁⠀⢀⣄⠠⠐⠀⡄⠀⢀⠜⠈⠀⠀⢀⠃⠀⠀⠎⠀⠘⡄⠀⠈⠀⠀⠀⠓⠂
⠀⠀⠐⠚⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⣿⣦⡶⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠑⠒⢀⠈⠁⠶⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⢣⠀⠀⠌⠀⢘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⡀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠙⣦⠀⠀⠠⣀⡀⢀⣀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠸⣧⠀⠀⣤⠟⢿⣄⠀⢹⡇⠀⠀⢀⣤⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⣿⢀⡾⠃⠀⠀⠹⣧⠈⣿⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⡀⠘⣧⣿⣿⠓⢶⣴⣖⡁⠹⣧⣿⣰⠏⠀⣀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⣿⣿⣇⣴⠟⠀⠈⠻⣦⣿⣿⡟⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠋⠛⠶⣤⡀⠈⢻⣿⣷⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠈⠻⣦⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡦⠀⠀⠀⠀⠀⠈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠇⠀⠀⠀⠀⠀⠀⠺⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣖⢲⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣄⡐⠟⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣎⣭⡛⠫⠭⠙⠉⠒⢄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠹⣆⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠉⠛⠢⢄⠠⣴⣵⡄
⠀⠀⠀⠀⠀⢸⡷⠲⠶⢶⠒⣿⠀⠀⠀⣀⣹⣄⠀⠀⠘⡿⠿⠿⠿⠶⣶⡶⠶⣤⣧⣀⡏⠉⠁⠒⢲⠀⠈⡆⣿⣿⡇
⠀⠀⠀⠀⠀⣸⣧⣀⣠⣸⠀⢋⡄⠀⣾⡟⠙⠻⣇⣀⣴⠗⣤⠀⠀⢀⣀⣈⣲⡀⠈⣿⣿⠀⠀⠀⢸⠀⠀⡇⣿⣿⡇
⠀⠀⠀⢠⡞⠉⠛⠾⠭⠭⠷⣿⠙⠓⠛⢻⡷⢾⠿⠿⣥⣾⣷⣤⡤⠴⠿⠿⠿⣿⣿⡿⠋⠀⠀⠀⢸⠀⠀⣟⣿⣿⡇
⠀⠀⠀⢸⠉⠓⠒⠒⠀⠐⢺⣿⣿⠛⠉⢹⡇⣧⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⡇⢀⠀⠀⠀⢸⠀⠀⣿⣿⣿⡇
⠀⠀⠀⢸⠀⡖⠲⠤⠤⠤⠞⣿⣻⡶⠶⣾⡇⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⡇⠀⠀⠀⠀⠈⠀⠀⠈⣿⣿⡇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⣤⣿⡛⣉⣃⡨⡳⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣃⡉⠉⠀⢀⣲⡄⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⣼⣽⣿⣷⣿⣿⣯⠨⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⠉⠉⠉⠉⠁⡇⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⡟⠻⠿⠿⠿⠿⠿⠂⢻⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⢯⣉⡓⣒⣒⣛⣩⡵⡀⠀⠀⢸⢿⣿⣇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⡇⠀⠰⢤⡰⡶⠶⠀⢸⣷⢄⣀⠀⠀⠀⠀⠀⢀⣷⢤⣄⣈⣉⣉⣁⣤⢴⡟⠋⠉⠙⠛⠻⠷⢤⣄⣀⡀
⠀⠀⠀⢸⠀⠧⢄⣀⣀⡇⠀⣴⠋⠀⠈⢲⠄⠀⣿⣿⠟⠻⠿⠿⠿⠿⠓⡏⠙⠓⠒⠶⠒⠒⠊⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢓
⠀⠀⠀⠘⠦⣄⣀⣀⣀⡇⠀⣈⡳⣤⡔⣁⠀⠀⣿⡟⠀⠠⡤⠒⠚⢿⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⡀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠈⠉⠉⠉⠉⢀⣠⡟⠧⢤⣀⣷⣷⡶⡟⠟⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⣤⡴⠊⠁⠀⢀⡤
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣿⣿⠿⠿⣿⣿⣿⠇⢿⠇⡀⠀⣠⣯⣧⠤⠂⡼⣿⠢⢤⣄⣀⣀⣠⠤⠖⢻⠿⣿⣿⣿⣿⣿⠀⢀⡤⠞⠉
⠀⠀⠀⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⠒⠊⠁⠘⠿⠾⠉⠉⡟⠒⠚⠋⢻⡀⠀⠀⠀⠀⠀⠀⣀⡟⠀⠀⠉⠉⠛⠻⠖⠉
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⠼⣧⣤⠴⠠⡄⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠚⠋⠁
⠀⠀⠀⣿⠤⠤⠤⠀⠐⠒⠒⠒⠒⠂⠉⢻⠀⠘⠶⠒⠊⠉⠁
⠀⠀⠀⠹⠴⠶⠒⠒⠒⠒⠒⠒⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠈⢻⣿⣿⡄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠻⣿⣷⡄⠀⠈⣿⣿⣷⠀⠀⢹⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣄⠀⠘⣿⣿⠄⠀⠸⣿⣿⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠙⠻⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡟⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠙⠻⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠟⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣠⣴⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀⣀⣠⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣶⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠿⠿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣀⣤⠖
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠿⠁
⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⣠⣤⣤⣤⣤⣴⣶⠞⠁
⣼⣿⣿⠿⠿⠟⠛⠉

⣤⣤⣴⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠁⠉⢉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠛⠛⠛⠛⠛⠳⠶⣤⣄⡀
⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄
⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠻⣆
⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠟⠋⠀⠀⠀⢹⡄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⣀⣤⡇
⣿⠀⠀⠀⣀⣀⣤⣤⠶⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⢹⡇
⣿⠗⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠞⠋⠁⠀⠀⠀⣸⠃
⢹⡄⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⢻⣄⣤⡴⠶⠞⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠶⠶⠶⠖⠛⣿⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠸⣧
⠀⠀⠀⠀⠀⠀⣼⠛⠛⠶⣤⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠘⣧⠀⢹⡆
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠲⣌⠻⣆⢀⡴⠞⠋⢉⡽⠛⠉⠛⣶⣄⠀⠸⣆⠀⢿⡀
⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⣈⠧⠹⠋⠀⠀⢠⡟⠁⠀⠀⣰⠏⠈⠳⣄⢻⡄⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄
⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠙⠃⠀⠀⠀⢠⡟⠀⠀⠀⣰⠏⠀⠀⠀⢹⣎⣿⡀⢘⣷⣶⡞⠛⠛⣿⠛⠶⣦⣀⠀⢀⣴⠟⣉⡁⠀⢿⡀
⠀⠀⠀⠀⡾⠁⠀⠐⠒⠛⠃⠀⠀⠀⠀⣼⠁⠀⠀⢠⡏⠀⠀⠀⠀⠐⠛⣽⣷⠟⠁⣿⠀⠀⠀⣿⠀⠀⠀⠙⢷⡟⣱⠞⠁⠀⠀⣼⠃
⠀⠀⠀⣸⠇⣴⣟⣋⡍⠀⣠⣦⠀⠀⢀⡏⠀⠀⠀⣼⡶⠞⠛⠛⠶⢦⣶⣾⠃⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣵⠖⠀⠀⢰⡏
⠀⠀⠀⠿⠤⢤⡭⠴⠶⠛⠉⢻⣄⠀⢸⡇⠀⠀⢀⡿⣇⠀⠀⠙⠛⠶⠌⠛⠀⠀⠀⣿⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠠⣤⣄⡀⠈⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣈⣧⠀⠀⢸⡇⠙⢦⡀⠀⠀⠒⠒⠂⠀⠀⠀⢻⡄⠀⠀⠘⣧⠀⠀⠀⠀⢀⡀⠂⠤⣤⣄⡀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠶⣼⡷⠶⠞⣿⠀⠀⣠⡶⠂⠀⠀⠀⠘⣷⠀⠀⠀⢹⣆⠀⠀⠀⣸⠻⣦⣘⠲⠾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢈⣡⠖⠀⣰⣆⡀⠀⠘⣧⡀⠀⠀⢻⣄⢀⡼⠃⠀⠀⠉⠙⠛⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⣏⡵⢋⡼⠋⠛⠛⠶⣤⣬⣷⣤⣤⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣭⠶⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢀⣈⣀⣀⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣴⣶⣾⡄⣶⣖⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠
⠀⠀⠀⠀⠀⢀⣶⣶⠘⠿⠋⣀⠉⢀⠈⠛⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⢐⡒⠀⠀⠀⠀⠀⠀⠀⢀⠀⢈⠥⠤⠖⠀⠠
⠀⠀⠈⠀⠐⢌⠻⢿⡿⠂⢸⣿⣿⠿⠃⠀⠀⠀⠀⠁⠀⠀⢁⠠⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠂⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⢈⠐⡤⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⡁⡂⠐⡀⢐⠀⣂⢡⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠤⠀⠅⠤⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⢌⢂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢰⡶⠄
⠀⠀⠀⠀⠀⢂⢐⠐⠂⠐⠀⡃⣈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠙⠋⠈⣡⣴
⠀⠀⠀⠀⠀⠀⠈⠀⠁⠁⠀⡀⠁⠠⠤⠄⠀⠀⠀⠈⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡿⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣀⠠⠆⠀⠤⡀⠣⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠀⠀⠀⢨⣤⠀⠀⠀⠀⠀⠀⠰⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡅⢠⣳⡶⠀⠁⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⢀⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠤⡅⠛⣀⠈⢀⠁⠀⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠢⠀⠀⠀⠀⠆⠀⠀⠲⠒⠀⠲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣴⣶⠦⠀⠀⠠⠀⠀⠀⠀⠃⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢀⣄⣸⣶⡟⣹⡏⢿⡇⠀⠠⠃⠀⠀⠀⢀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⠂⠀⢸⡶⢋⣿⠛⣿⡍⢛⠂⠛⠃⠊⠁⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠸⠿⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠁⠒⠤⢤⡀⠀⣀⣀⣰⡿⢯⡙⠃⢀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠶⠞⠀⠀⠀⠁⠀⡀⠀⠈⠜⢇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠂⠉⡆⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⢀⠃⠀⢠⡇⡟⠁⠀⠀⠀⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⢀⡴⠫⣤⣤⡀⠀⠀⠀⡎⠀⠀⣸⠁⠩⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠩⠀⠀⠈⠁⠂⢀⣀⣼⡄⠀⠒⠋⠀⠀⠈⠁⡄⠀⠀⠀⢀⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠑⠀⠀⠀⠁⠀⢀⡰⣄⠀⣀⡔⣇⠀⠀⠀⠈⠀⡐
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡤⠈⠁⢏⠲⢶⡶⠸⢇⠀⣀⡀⠀⠸⠋⠙⠉⠀⡞⠓⠀⠣⠔⠊
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⡁⠀⢀⣨⡷⡾⠀⠀⠀⢢⡿⠿⠶⣾⣦⣶⠖⠛⠛⠂⠀⠀⠄
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠢⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠧⣤⠼⠷⣶⣿⡏⠀⠈⢤⠞
⠀⠀⠀⡀⣠⠇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢤⠀⠀⢀⣾⣶⣿⢶⡄⠀⠈⠓⡄⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠈⠀⠀⠀⡀⠈⠀⠀⢻⠀⠒⠀⠄⠤⠄⠑⠒⠊⠘⠉⡉⠹⣭⣠⡦⠴⠉⠒⠈⠓⠀⠈⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⡨⠀⣠⡆⠀⠀⠂⠈⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡠⠖
⠀⠀⠀⡠⠄⠀⠀⠐⢾⣆⠀⠀⠀⠀⠀⠀⡘⠁⠀⠃⠁⠀⠀⠈⣀⠄⠀⠀⠠⠄⠀⠀⠀⠀⠠⠊⠀⢄⠀⠈
⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠜⠋⠔⠀⠀⡀⠀⡠⠀⠀⠀⠀⠀⠀⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⣸⠶⠞⠋⡄⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢧⣸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⢀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢠⡏⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣸⠃⠀⠀⠀⠀⠀⠈⢿⣦⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣄⣀⣸⣿⠀⢹⠀⢀⡄⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠛⠉⠉⠉⠉⠉⠉⠻⠀⠈⠀⡾⠀⠀⠀⠀⣀⣠⠄⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠴⠛⠉⠀⣠⡤⠶⠚⠛⣻⡆⠀⠈⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⣀⠤⠴⠚⠉⣀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⡀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀⣀⣤⣤⡶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠘⠁⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠐⣷⡀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣆⠀⠀⠀⠀⠀⠀⠙⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⡿
⠀⠀⠀⢀⣠⣴⡶⠿⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⢀⣴⠟⠋⠀⢀⣤⠴⠖⠒⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⢶⣤⣤⣤⣤⣴⠟
⢠⡟⠁⠀⠀⠜⠉⠀⠀⠀⠀⠶⠋⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣽⠟⠁
⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠰⣶⣤⣤⣤⣤⣴⡶⠾⠛⠉
⠀⠈⠻⢷⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⣸⠁⢀⡆⠀⠀⢿⡇
⠀⠀⠀⠀⠉⠉⠙⠛⠛⠿⣦⡀⠀⠀⠋⠀⣼⠁⢀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⢀⡏⠀⣼⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠘⣇⣰⠏⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣄⠀⠈⠁⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣀⣾⠏
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⢀⠠⠀⠒⠒⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⡘⠢⡀⢀⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠰⠀⠀⠸⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠲⢬⠂⠀⣀⠀⠀⡆⠀⠰⠁⠈⣷⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠐⢄⠀⠀⠰⣯⡆⢠⠀⠠⠁⠀⠀⠘⡇⠀⠀⠀⠀⠆⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠢⠀⠀⠻⣷⣈⠀⠃⠀⠀⠀⠀⢥⠀⠀⠙⢺⣦⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠀⠀⠱⡀⠘⣿⡏⣨⠀⢠⠎⠀⠀⢸⡇⠀⠀⡆⠘⣧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠀⠀⠀⠈⣉⣛⣿⣦⣄⠀⣹⣆⣿⡇⣿⣷⣁⡄⠤⠤⢴⣧⣀⣰⠀⡔⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠁⠀⠀⠀⢳⢶⡆⠈⠀⠀⢻⡍⠟⣯⢹⡟⠘⡿⣫⣤⡀⠀⢀⠐⣓⣨⢭⢾⣄⠀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣀⣠⣤⣄⣐⠁⠀⠈⠓⡄⠀⠘⣧⠀⡿⠀⠃⣻⡇⠸⣧⠙⢀⠖⢸⣧⣾⠋⠀⠉⠣⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠈⠉⠛⢶⣄⠀⠘⢦⠶⢿⡆⠁⠘⣶⡿⠀⠰⢋⠀⢳⠀⣸⣿⠃⠀⠀⠀⠂⠁⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠿⣤⡾⢶⠼⣷⣷⠀⠈⡵⠁⠀⢾⡆⢸⠀⣿⣥⣊⠁⠀⣀⣀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⠸⣧⡌⠂⣿⠈⣷⠀⣧⠂⠀⠸⢣⡞⢸⡿⠋⠀⠀⠀⠀⠙⢷⣦⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠠⢀⣀⣀⣠⣧⡏⢻⣄⣿⣠⠿⠘⢃⣋⣤⢢⣾⣧⣿⣶⡶⣄⣀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠓⠇⠀⠉⠀⠉⠉⡟⢷⣿⣾⣧⣴⠂⠘⠉⡋⡜⢠⠋⠠⠀⠀⢿⡉⠛⠂⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠁⠀⠀⠀⠀⣷⠃⠙⢿⡟⠋⠀⡄⠰⠁⣀⠏⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠐⢆⣰⠉⡿⢀⡟⢿⠁⡀⢌⠀⠀⠀⣼⡤⠔⠺⠗⠲⢾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⠘⢻⣾⡇⡼⠁⣿⡞⢃⣼⠀⠀⢸⡋⠀⠀⠘⠀⠀⠀⡿⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⡉⠳⣼⣿⣧⢀⣤⡏⠀⠈⠙⠀⠷⡿⢻⡆⠀⠁⠀⠀⠀⡇⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡆⠉⠁⠉⠐⠨⢿⣿⣼⠹⡇⠀⠠⣾⠀⣸⠇⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⢻⡋⠰⣷⠀⢀⡾⢀⣿⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠈⣇⠀⣿⠀⠀⡇⠸⡏⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢻⡀⢿⡇⠰⡟⢸⡇⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢸⣇⢠⢷⣼⡇⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢹⣬⣿⢀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣼⣻⣿⠈⣿⣿⣷⡔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⢹⣿⣿⢉⣹⣿⣿⠋⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣦⣾⣿⣿⣿⣇⣿⣿⣿⣿⡬⣵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠸⠿⠷⣶⣽⣏⣿⠍⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⢠⣾⣷⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⣤⠖
⠀⠀⠀⠀⢀⣥⠼⢻⡿⣮⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣤⢀⠈⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⡄⠀⠀⠀⠐⢻⣿⠟⠉⠉
⠀⠀⠀⠘⠆⠟⢠⣾⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⡀⠀⠀⠄⠀⠀⠀⢹⡫⢿⣿⡇⢀⣴⠏⠀⠀⠁
⠀⠀⠀⠰⡐⡦⠀⠙⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⠀⠀⠀⠀⢃⣾⣿⣷⣼⣿⣄⣠⣄⣀⠠
⠀⠀⠀⠀⢋⣀⣤⣤⠀⡀⠀⠀⠀⠀⠀⢀⠀⠶⠆⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠛⡿⢏⡉⠉⠉⠉⠚⠛⠀⢠
⠀⠀⠀⠀⠆⠈⢳⠉⠶⠃⠀⠀⠀⠄⣤⣶⠇⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⢤⡀⠀⠀⠀⠀⠃⣤⣹⣤⣠⡄⠰⠆⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠀⢲⡀⡄⠀⠀⠀⠀⠀⠀⠀⠐⠋⣰⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⢠⣶⢶⣶⡀⠀⠀⠀⠀⢀⡞⠁⠀⠸⠀⠂⠀⣀⡀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠈⠁⠀⠀⠀⠚⠃⠀⠀⠀⠀⠀⠀⠉⣿⠂⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠘⠃⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡄⠀⢠⠵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣠⣦⠀⣰⣶
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀⢸⡇⠀⠀⢀⣠⣤⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢿⠆⠈⢡⡶⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡶⠀⠸⣿⣿⣷⡄⡀⠃⠀⠀⠘⢿⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⡄⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠀⠀⠀⠀⠀⠀⠹⢿⣦⡀⠀⠀⠀⠀⣠⡴⠃⠀⠀⣸⡶⠃⠙⠇
⠀⠀⠀⠀⠀⠀⠐⠀⠂⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢻⣿⣿⢹⣿⣿⡇⠘⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⣿⠄⠀⠀⠀⠉⠐⢦⡄⡀⢈⣀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠉⢠⣼⡿⢹⠀⠐⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀⠂⠁⢀⠈⢁⡀⢉⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠻⢿⡇⠀⠀⣿⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢿⣵⠿⠂⠤⠿⠟⠉⠁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⣷⠀⠀⣶⣦⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⢀⠀⠈⠠⠖⠈⠿⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⢷⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣰⠟⠁⠀⠀⠀⠀⠰⢾⡷⠀⡈⠁⠛
⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠄⠘⠀⠀⠀⡄⠀⣾⠀⠀⠀⠀⡿⠋⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠄⠀⢠⣤⡄
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠎⢹⠏⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢰⡇⠀⢹⠀⢀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠲⠶⠛⣉⠁⢀⡀
⠀⠀⠀⠀⠀⢗⠀⠀⠀⠀⠘⠀⠼⠄⣸⠆⠈⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⣸⠀⠀⠘⠄⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡚⢛⣿⣿⠟
⠀⠀⠀⠀⢠⡯⠙⠁⠀⠀⠀⢠⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⢀⠀⠈⠉⠿⠉⣠⡤
⠀⠀⠀⠀⠼⠖⠂⠀⠀⠀⠀⢠⣠⡄⢸⠀⢸⠀⠀⠀⣷⠈⢳⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡍⣲⠆⠘⠀⠀⢀⡀⠀⣩⣴
⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⢾⠀⣿⣧⣸⠀⡆⠀⠀⠀⢩⠀⠈⠀⠀⠀⠀⡟⠀⠀⠀⢘⡇⠀⠀⠀⠀⣶⣶⣤⣄⠀⠀⠀⠀⠀⠈⣿⠿⠆⠀⠀⠀⠀⠙⣾⠟⡁
⠀⠀⠀⠀⣙⠀⠀⠀⠀⠀⠘⣿⣿⣿⠇⢠⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣾⡇⠀⠀⠀⠀⢸⡿⣿⣷⢂⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⢹⣿⣿⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢣⡘⠁⠀⠀⠀⠀⠀⢸⠟⠻⣿⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠇⠀⠀⠀⡀⠸⠇⡀⠀⠀⠀⠀⠀⠀⠛⠖⡉⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠠⠄⠷⣤⠀⡀⠀⠰⠀⠀⠀⢸⠀⠀⠁⠀⡄⠀⠀⠀⡀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣀⣀⢠⣶⣏⠉⠃⠀⠀⠀⠀⠀⢨⡇⢰⠀⠈⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢾⠿⡏⢰⣷⣷⣔⣀⠀⠀⠀⠀⠀⢺⡇⠘⠀⠀⠀⠀⠀⠀⢀⡛⢗⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡒⠛⠾⠟⢻⣿⣯⣀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⣀⠈⢱⣎⣹⣇⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⡟⠉⡄⠉⣶⣦⡀⡿⠁⠜⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡀⠀⠀⣹⡆⠰⢃⠀⠠⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠘⠂⠀⠀⠀⠀⠃⠀⠁⠸⢿⣈⣄⣷⠞⠛⠀⠀⠀⠘⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⠷⠒⠞⠳⠶⣦⡀
⠀⠀⠀⠀⠀⢀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⣄⣀⣏⢈⣀⠀⠚⠀⠀⠀⠀⣿⠀⠀⠀⣀⢠⣴⣾⣷⠒⢠⣄⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠈⠉⠉⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠙⣏⠛⠃⣴⠆⠀⠀⠀⡟⠀⠀⠀⢫⣶⣯⣽⡏⠐⢀⡽⠻⡇⣸⠂⠀⠀⠀⠀⠀⣀
⠀⠀⠐⠃⠀⠀⠀⠀⠀⠄⠀⢀⠀⠄⠀⣀⠄⠀⠀⠀⠀⠀⠘⢶⡸⠇⠀⠀⠀⠀⠃⠀⠀⠀⢣⣾⡿⡏⢙⠀⠹⠀⠀⠀⢉⠀⠀⠀⠀⠀⠠⠿⡿⠃
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠃⣴⠃⠀⠀⠀⠀⠀⠁⠢⠀⠀⠀⠀⠀⠁⢄⠀⠀⠀⠀⠀⠀⠀⠀⠿⣹⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⢀⠀⡰
⠀⠀⠀⠀⠀⠀⠉⠀⠐⢦⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠜⠓⠠⠰⠇⠺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⡀⣀⣀⡀⢀⣀⣀⣀⡀⣀⠀⢀⡀⠀⣀⣀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⣀⡀⣀⠀⢀⣀⢀⣀⢀⣀⣀⣀⣀⡀⢀⣀⠀⣀
⣿⣿⡟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣜⣋⣻⣧⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡍⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠈⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠂⠠⠀⢠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⢀⡠⠉⠀⠄⠀⠁⡀⣸⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⠀⠀⠀⠀⠀⠠⠄⠨⠀⠀⠀⠘⡶⠶⠤⠞⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠈⡑⠒⠐⠂⠁⠉⠒⠆
⠀⠀⠀⠀⢐⠰⠖⠂⠀⠀⠀⢤⠆⠂⠂⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠐⢄⠀⠀⠲⣄⡀⢂⠀⠀⠴⠊⠐⠀⡀⠀⠂⠀⠀⠀⠂⡁⠐⠇⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⢄⠀⢶⡀⠀⠀⠀⢈⡽⢣⠀⠀⠂⠠⠀⠈⠃⢄⠀⠀⠔⠊⠀⠀⠀⠂⠀⠀⠐⠁⠀⣀⠄⠀⠀⠀⠒
⠀⠀⠀⠀⠘⡎⠀⠁⠀⠀⠦⠞⢁⠇⠰⡀⠠⠀⠀⠠⠀⠀⠀⠆⠀⠀⠀⠄⠀⠀⠀⠆⠀⠀⠰⠁⡀⠀⠐⠀⠀⠀⠓
⠀⠀⠀⢀⡠⠁⠀⠦⡘⢦⡐⡐⠂⢤⡀⡸⢁⢠⠀⠀⠀⠠⠀⠀⠀⠤⠀⠀⠀⠄⠀⠀⠠⠄⠀⠀⠰⠀⠀⠀⠰
⠀⢀⠀⠠⠂⢰⢦⣄⠹⡬⣯⣄⣠⡿⢱⢡⣿⢡⢒⣤⠀⠀⠀⢠⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⠠⠄⠀⠀⠠⠀⠀⠀⢠
⠀⠠⠀⠀⠀⠀⠀⠘⣏⣐⠪⢭⣿⢿⢷⣾⣧⣇⣾⠇⠠⢀⠀⡀⠀⣀⠠⢤⠤⡄⠀⠀⠀⠄⠀⠀⢤⡀⠀⠀⠠⠀⠁
⠀⠀⡤⠀⠀⢀⠴⠚⠩⣛⠻⣿⣿⣯⣾⡟⡿⣿⡿⠷⣶⡖⣀⡉⠤⢀⠄⣨⠖⠂⠀⡀⠀⢀⠐⠁⠀⠈⠂⢀⠀⠀⠠
⡀⠊⠀⠀⠀⠁⠀⢠⠿⣶⣼⣟⣺⣽⣿⣻⣿⣿⣬⣍⡒⣭⣤⣄⡀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠
⠃⠰⠀⠀⠀⠁⠀⠀⢂⢠⡿⢹⢻⣿⢷⣧⣿⣓⠨⠍⣻⠉⠀⠐⠁⠠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢁⠃⣾⢃⢇⣾⠏⠙⣻⠂⠈⡇⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠚⠋⠀⠁⠀⠢⠀⠁⠀⢀⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡠
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡟⣩⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠈⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠈⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣿⣼⠏⡠⢼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⠁⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⢃⡈⣁⠐⢹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣻⠟⠉⡹⠛⢻⣯⣊⡔⣡⡼⠃⠀⠀⠀⢀⡠⠖⠋⠉⠉⠉⠙⢀⡀⠀⠀⢀⡸⠃⠀⠀⠀⢀⡤⡖⠛⠩⠙⠋⠽⢰⡀⡄⢨⣁⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⡴⠃⣠⠞⠀⢀⠜⡏⠉⠉⠉⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠀⠀⠀⠀⣰⠛⠂⠆⣉⠰⠀⡁⠐⢨⡏⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢰⡟⠀⡼⠁⡴⠃⠀⡴⠋⣠⡇⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⣰⠧⣈⠐⠀⠄⢐⠀⠄⡃⢴⠇
⠀⠀⠀⠀⠀⠀⣿⠀⡼⠁⡼⠁⢀⠞⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⡟⡂⠤⢈⡁⠆⢨⠐⠂⣤⠎
⠀⠀⡔⣋⡽⣻⢛⣶⣃⣸⠁⢠⣏⣠⠟⠁⠀⠀⠀⢀⡔⠋⠉⠉⠙⢂⣀⡀⠀⠀⣀⣠⠞⠁⠀⠀⠀⢀⣴⠋⠛⢩⠙⣅⣂⣠⠀⣁⣰⣬⠟⠁
⠀⠈⠯⣥⠞⠁⡼⢸⡏⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠦⣤⠀⠀⠀⠀⡏⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠘⠧⣤⠁⠰⠀⢆⡏⠉⠉⠉⠁
⠀⠀⠀⠘⠤⡼⢁⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⡄⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢭⡄⡃⡻
⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⢀⠔⠂⠀⠀⠀⠀⠺⣇⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⢸⠠⡀⠀⠀⠀⠠⣺⠆⠿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠸⠀⠉⠛⠷⠿⠛⠁⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠐⠻⣦⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⢶⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠃⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣐⠲⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠠⣿⠀⠀⣰⠋⠃⠒⠂⠀⠀⠀⠀⠀⠀⢶⣤⣀⡀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠘⠁⢰⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠙⠛⠿⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⣐⣤⢀⣽⠃⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠸⠁⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠉⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢂⠀⠀⠀⠤⠀⣶⣾⣿⣿⣿⣤⡖
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⢷⣄⠀⠐⢇⢹⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄
⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠙⢿⡄⠀⡐⢛⡻⣿⣿⣿⠧
⠀⠀⠀⠀⠀⠀⠀⠟⢤⡀⠀⠀⠀⠈⠃⠀⠈⠀⠁⠘⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠹⢦⠀⠀⠀⠠⠐⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠤⠀⠄⠠⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡻⠀⠀⠀⣠⣤⣦⣴⣤⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠀⠁
⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡴⢀
⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠐⠳⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⡀
⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠛⢷⡄⠀⠀⠀⠀⠀⠀⠀⠙⠷⠃⠈⠿⠟⠷⣤⡄
⠀⠀⠀⠀⢸⠛⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⣀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣄⠀⣸⡟⠛⢢⡤
⠀⠀⠀⠀⠀⠠⠙⢧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⣷⣠⠀⡀⠀⠀⠀⠀⠀⠌⣡⠈⠛⠀⢀⠄⠂⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⡌⠻⡇⢠⠀⠆⠀⠀⠀⢀⡄⢀⢸⡟⠃⠐⠛⢿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠹⡆⠀⠀⠀⠀⠀⢾⣠⣿⠦⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠀⠀⠀⣀⠶⣿⢶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⠛⣿⡏⠠⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠹⣏⢛⣛⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣘⣠⡄⠀⠀⣤⠀⡀⠀⠀⠀⠀⠐⠋⢈⡯⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣂⠀⣄⡀⠀⠀⠀⠂⠀⠀⢸⣆⢠⡿⠀⠇⠀⠀⠀⠂⢀⠀⣠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⡾⣟⣿⣿⣿⣷⣦⠄⠀⠀⠀⠀⢠⣿⡼⠃⠀⠀⠀⠀⠀⠀⣼⣟⣧⣶⣶⣠⢶⠟⠠⠒
⠀⠀⠀⠀⠀⠀⣀⠀⠀⢬⣿⢰⣿⣿⣿⣿⣿⣾⣷⣇⡆⠉⠀⠀⠀⠁⠀⠀⠀⣀⠂⠀⠀⠛⠿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠹⠦⠀⢸⡞⣿⣿⣿⣿⣿⣿⣿⣿⣦⣈⠀⠀⠀⠀⣀⠀⠘⢧⣄⣁⡀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⢽⣿⣿⣿⡋⢨⣿⣯⡀⠀⠀⠀⠀⠀⡀⠐⠃⠈⢻⣿⡅⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠉⠈⠁⠉⠀⠈⠉⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐⠋⠹⣿⡀⠀⠀⠐⢂⠀⢠⣽⠟⣻⣷⣿⢦⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠠⣼⡟⣾⣿⣿⣿⣷⣌⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄⣀⡀⠀⠀⠀⠐⠂⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠱⠀⠀⠀⠀⠀⠼⡿⠛⠿⣿⣿⣿⣿⢻⣷
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠚⠛⡻⣦⣤⣤⣄⠀⠀⠠⣤⣤⡤⠡⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠌⣀⠀⠋⠀⠛⠁⠀⠀⠐⢶⡿⢛⣿⢉⣼⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡠⠴⣿⣿⡣⠙⢧⣄⠐⠈⢻⣿⣷⣤⡀⠀⠀⠀⢀⣤⠞⠛⠛⣰⠛⠃⡀⣀⡠⠞⠀⠄⠤⣀⣀⢈⣙⠂⠀⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡳⠛⡀⠘⠃⠀⠀⢘⠇⠘⢿⣶⡙⢠⡀⠈⠀⠇⠄⠀⠚⠀⠀⢐⠛⢧⡄⠀⠀⠀⡉⠉⠈⠛⠲⣦⠈⠁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠟⠗⠀⠁⠀⠀⠀⠀⠸⠀⠀⠜⣿⡷⣀⠁⠀⢀⣠⡴⠶⣶⣬⣄⠀⠀⠀⠀⠀⠄⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⡉⠂⠀⠀⠉⠀⠈⠉⡉⠿⠆⠀⠶⠶⣶⣗⡀⠛⠶⡾⠹⠒⣶⠂⠡⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠸⠛⣷⡄⠀⠀⠀⠄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠻⠙⠿⡟⠙⡿⠻⣟⠿⢛⣻⠟⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣿⣶⣷⣾⣷⣶⣿⣶⣾⣿⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠰⣿⣿⠋⠁⢀⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⢻⡆⠀⠛⠛⠀⠀⣾⠃⡠⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣰⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠙⠿⠂⠀⠀⠀⠀⠀⠈⠈⠐⠒⠦⣄⠦⣄⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⢠⠖⣤⣱⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⢸⠄⠀⠙⣿⣧⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢼⡀⠀⠀⠿⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⢰⣿⠀⠀⠆⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠲⠾⠛⠒⠒⠦⢄⣹⡆⠀⠀⣠⣾⣿⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⢰⣾⡿⣿⠤⢴⠞⠛⠉⠉⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⡴⢷⣄⣼⣦⡀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠿⠛⠛⠿⠁⡠⠟⠛⠻⣷⣄⠀⠀⠀⠀⠀⠀⠈⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠋⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠈⠻⣧⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⣀⠀⣀⡀⠀⠀⡆⠀⠁⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠐⢻⡄⠘⣿⠋⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣧⠀⢻⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠈⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⢸⡟⠁⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡄⢸⡇⠀⠀⠀⠀⠀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠰⣿⣿⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠉⠉⢉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⢶⡀⠆⠠⢀
⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠘⠳⢦⠀⠀⠀⠀⠀⠀⢸⠇⠀⡀⠂⢼⡇⠀⠀⠀⠀⠀⡀⠀⠀⢀⡩⠄⠚⠻⠇⠀⠄⣀⠀⠀⠈⠂⠁⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠤⠀⢏⠀⠀⠀⠀⣹⣟⡀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣾⢡⣶⠋⠀⠀⠀⣀⣀⠐⢾⡏⠁⠀⠀⠀⠀⣸⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠰⠋⠀⠀⠃⠀⠀⠀⠀⠐⠰⣤⠖⠀⠀⠀⠀⠀⠀⡄⢸⠹⣯⠀⠀⠮⠴⠊⠀⠀⠀⠀⠀⠀⢀⡜⠦⡈⠺⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣤⠶⣉⠀⡩⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡙⠦⡈⠲⢌⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡤⠠⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⢦⡈⢳⣤⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⢁⠀⠀⠠⣿⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀
⠀⠀⠀⠀⠠⣿⣦⠀⠀⠀⠀⡀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⣠⠄⣿⠃⢸⡟⠀⣿⠀⡄⡄⠀⠁⡀
⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠈⢭⣿⠀⣿⡀⢸⣷⠀⣿⣀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠙⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠐⠀⠀⠀⠉⠃⠻⠇⠈⠿⠄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⡤⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣁⡴⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⣠⠁⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⢀⡀⣀⡠⠀⡀⠀⠀⠀⠀⠀⢀⡤⠚⠿⢿⣷⣶⣶⣐
⠀⠀⠀⠀⠀⠀⠀⡴⠊⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣌⠀⡴⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⢀⡛⣿⣿⠉
⠀⠀⠀⠀⠀⠀⣨⣤⣴⠏⠀⠀⠀⠀⠀⠀⠁⠐⠈⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠘⠿⢇⡀⠀⠀⠀⠀⢀⡀⠀⢀⡴⠋⠄⠸⠟
⠀⠀⠀⠀⠀⠰⢋⣀⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⣈⡁⢩⡌⢈⠛⡄⠀⠀⠀⠀⠁⠀⠉⣤⡄⠸
⠀⠀⠀⠀⠀⠿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠐⡍⢁⣌⠀⢙⡁⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠌⡅⢨⡦⠁⠒

⣿⡟⣛⡿⠿⠿⠿⠿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⢦⠀⠰⢄⣀⡀⠀⠀⠂⠸⡆⠀⣾⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⢸⠇⠀⢿⡑⡈⠁⢒⢨⣿⠃⠀⠘⠫⣼⣭⣭⡻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣶⡾⠇⠀⠀⠁⠙⢻⠋⠋⠀⠀⠀⢠⣶⣯⣿⣿⡄
⠀⠀⠀⠀⠀⢺⠤⢄⡀⠈⠈⠀⠂⠀⠀⠀⠀⢰⡆⠀⢰⣧⡀⠀⠉⠙⡏⠛⠁⠀⡤⠓⠀⠀⠀⠀⣤⣀⣤⣀
⠀⠀⠀⠀⠀⠐⠀⣜⡳⣶⣤⢀⠀⠀⠀⠀⠀⠀⡇⢀⣺⣾⣧⡄⠀⠠⠀⠀⢤⠒⠻⣀⣀⣻⠿⢟⠟⣿⣯⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢥⠈⠙⠿⣧⠀⣄⠀⠀⠆⢳⠀⠈⠇⣷⣆⠀⣈⠰⡖⠻⢇⣴⠌⣋⡵⣂⠱⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠐⡮⣷⣝⠿⠁⠀⢻⣦⡄⠈⠿⣗⣠⡟⠉⣡⣾⣿⣿⠯⠅⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠔⠀⠛⠀⠿⢿⣦⢻⡧⠀⢚⣿⣿⣴⠿⠿⡙⣭⣿⡶⣯⣤⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⠀⠀⡀⠀⠠⢂⣀⣶⣮⣝⣿⡂⢨⡿⢟⣾⣭⡎⠄⠩⠜⠙⠀⠙⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⣈⣍⠉⣲⣿⣿⡏⠃⠀⣿⣿⣿⣿⣿⠾⠶⠶⠤⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠙⠛⠓⠛⠛⠋⠙⠛⠃⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢯⢣⠈⢻⣄⠀⣀⡠⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⢁⡀⠀⠡⡑⣄⠹⡏⠀⠀⠀⠀⠀⠀⡴⣚⣛⠻⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠛⠁⠀⠈⠁⠀⠀⠈⠪⡴⠁⠀⠀⠀⠀⠀⠈⠘⢐⠐⠁⠀⢠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣧⡀⠋⠀⠀⡄⠀⠀⠘⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠥⢐⢫⠷⠶⠮⡙⢆
⠀⠀⠀⠀⠀⠀⣀⣀⠸⣧⠈⢔⠄⠀⡀⠀⠀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠠⡀⡀⡼⠄
⠀⠀⠀⠀⠰⠍⠀⠀⠀⠈⢷⡀⠢⠂⣋⠔⠊⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⣄⣀⡀⠀⠒⣬⡥⠞⠁
⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⢀⠟⠓⠉⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠘⠛⠖⠀⠄⠠⠄⢀⢀⣀⣀⠠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⢀⢀⠐⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⢀⠡⡐⣂⠀⠶⢦⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⢁⠀⠀⠀⠀⠀⡀⠄⠂
⠀⠀⠀⠀⠰⠚⠂⠃⠀⠀⠀⡀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⠂⠀⠀⠠⠖⠥⣀
⠀⠀⠀⠀⠀⣀⠀⠼⠖⠒⠈⢁⠀⠀⢸⡆⠀⠀⠀⠑⢄⠀⠀⠀⡠⠊⠀⠀⠀⠠⠁⠀⠀⠀⠉⠒⢤
⠀⠀⠀⠀⠀⠁⠀⢀⣠⠠⠤⠐⣆⠀⠩⣦⠀⠀⠀⠀⠀⠁⠲⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠈⠧⠀⠠⠔⠁⠒⠁⢀⠀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⣀⠀⠄⠐⠀⠈⠀⠀⠀⠘⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⠄⠄⠂⠈⠂⠙⠀⠀⠀⠀⠀⠀⠀⠀⠠⣅⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⣀⡠⢁⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡔⢋⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⣀⠠⠄⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⡠⠀⠒⠒⣒⠒⠬⢥⣄⠀⢀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣇⠴⡴⢶⠀⣦⠽⠍⠛⠀⠒⠢⢨⣽⢡⣆⠨⣌⡑⠠⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⣡⡪⠴⠊⠂⠙⢬⠑⠤⠤⠀⠀⠀⠀⠀⢦⠙⣈⠉⠀⠓⢌⡙⠦⠀⠜
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⡐⣩⠈⠀⠀⠀⠀⠀⠀⠁⠀⠒⠁⠀⠀⠀⠀⠈⠢⢉⠆⠀⠀⠀⠑⢥⣹⠆⠠⢄
⠀⠀⠀⠀⠀⢨⠀⡀⠞⣦⠈⠇⠠⠒⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠠⢮⡀⡅⠳⡀
⠀⠀⠀⠀⠀⡬⡖⢡⠄⠁⠈⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠸⡢⡰⢄
⠀⠀⠀⢀⡔⢡⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠲⡘⣍⠉⢢
⠀⠀⠀⠌⠂⢋⡆⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣌⣦⠀⠁
⠀⠀⠀⠐⣌⠛⠖⡈⡡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⣮⡤⠈⢂
⠀⢎⠐⡴⠋⡛⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡆⠀⠗⡇⢸
⠀⠈⠃⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠛⢸
⠀⠀⢸⢰⡦⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⡗⢦
⠀⠀⢸⠀⡧⣆⢀⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠠⣇⡼⠡⡀⠂
⠀⠀⠀⠊⡚⣋⢇⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣱⠔⣦⠄⠑
⠀⠀⠀⡀⡘⠏⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢃⠆⣢
⠀⠀⠈⠠⣌⣣⠙⡆⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡛⡠⠃
⠀⠀⠀⠀⠀⠐⡁⢿⡠⡀⠘⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡄⢀⡠⠚⡰⢏⠁
⠀⠀⠀⠀⠀⠀⠙⡤⠑⢹⠞⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠖⠁⡶⠚⡠⠋⠅⠸
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠺⡍⢢⡀⠀⠀⠀⢍⡑⢄⠀⠀⠀⠀⠀⠠⠠⠀⡀⠀⠀⠀⠀⠀⠠⠊⢁⡟⠅⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢘⢢⡌⠢⣄⠄⣀⣡⡝⡆⠀⠀⠀⠀⠀⠛⠠⡍⣣⡤⢀⠖⠾⢁⡾⠗⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠢⣌⠉⠀⠏⣸⣋⠁⠲⠤⠠⣔⣠⠫⠓⠸⠖⠳⠃⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠄⠉⣹⠫⠆⠠⠴⠎⠀⠀⠀⠑⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉
⣟⣿⣿⣿⣛⣛⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣿⣛⣻⣛⣛⣿⣿⣻⣿⣛⣻

⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠛⠛⠛⠛⠀⠀⠀⣈⣁⠀⢀⠀⣤⣬⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢛⣷⡿⢫⠁
⠀⠀⠀⠀⠀⠀⠤⢶⣶⡶⠓⢀⣈⣻⡜⢎⢿⣦⠆⠀⠀⠀⠀⢀⠀⣠⣴⡿⢋⣴⡏⣠⣀⣀⡤
⠀⠀⠀⠀⠂⣾⡿⠠⣤⣤⣾⡿⣿⣻⣿⣎⢧⠝⠀⠀⠀⠀⣀⠊⠀⠝⠁⣴⠟⢻⣿⡈⣀⣈⣄
⠀⠀⠀⠀⠀⠇⣰⠚⣫⣽⠿⠟⠛⠉⠀⠈⠁⠀⠀⠀⠀⠉⠉⠛⠳⢦⣤⣧⣤⠛⠉⢛⣻⣿⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠛⠛⠋⠁
⠀⠀⠀⠀⢀⣀⣐⣀⣒⣖⣂⣀⣒⣰⣆⣀⣐⣒⣲⣶⣖⣒⣒⣂⣰⣖⡒⠒⠒⢲⣒⣂⣀⣖⣒⡀
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠈⠀⠉⠁⠀⠉⠉⠉⠉⠈⠁
⠀⠀⠀⠠⠀⠀⢀⠀⣂⡀⠀⠀⠀⠀⠀⣀⢀⣀⣀⠀⠀⢀⡀⣀⣀⢀⣀⣀⢀⡀⢀⠀⠀⢀⢐
⠀⠀⠀⢰⣦⣴⣼⡁⣯⣤⣴⣤⣦⣶⣿⣿⣾⣿⣵⣾⣿⣩⣴⣶⣠⣤⣄⣧⣬⣧⢸⣴⣮⣤
⠀⠀⠀⠀⢻⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠮⣿⣿⣿⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠶⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠶⠲⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣤⣤⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⠴⠤⠴⠴⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠰⠶⠶⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡅⠶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⣀⣀⣀⣀⣀⠀⣿⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⠀⣿⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⠀⣿⠐⠛⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢋⣠⣾⣿⣿⡿⣿⣿⣿⣦⣌⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⣡⣴⣿⣿⣿⢿⣿⣿⣷⣤⡙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣴⣿⣿⣿⣿⣍⡁⢉⣽⣿⡿⣿⣷⡌⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢡⣾⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣦⠙⣦
⠀⠀⠀⠀⠀⠀⠀⢸⠇⣼⣿⣏⠃⠸⣿⣿⡇⣿⣿⣿⠆⢋⣽⣿⡄⢻⡄⠀⠀⠀⠀⠀⠀⢀⡟⢠⣿⣯⡙⠻⣿⠟⠛⠛⠻⢿⡿⢛⣹⣿⣧⠸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣧⣶⣶⣬⡙⠃⠛⣉⣴⣶⣦⣽⣿⣷⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⣾⣿⣿⣿⣷⠃⠀⠀⠀⠀⠈⣷⣿⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⡛⠻⠟⢋⣡⡄⣤⣍⠛⠿⠟⣻⣿⡿⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⢿⣿⣿⣿⡿⣆⠀⠀⠀⠀⢀⡿⣿⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠹⣿⣏⡄⢸⣿⣿⡇⣿⣿⣿⡆⣬⣽⣿⠃⣾⠁⠀⠀⠀⠀⠀⠀⠈⣷⠘⣿⣯⣥⣶⣿⣶⣤⢤⣴⣿⣷⣦⣹⣿⠏⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠙⢿⣿⣿⣿⣋⡄⣈⣹⣿⣿⣿⡿⢁⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡈⢿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⡿⠋⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡙⠻⠿⣿⣷⣿⡿⠿⠛⣁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣈⠛⠿⢿⣿⣾⡿⠿⠟⣋⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⠶⠶⠶⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠶⠶⠶⠶⠛⠉

⠀⠀⠀⠀⠀⣀⠤⠀⠀⡤⣀⢐⠉⠂⠲⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠠⣀⡀
⠀⣅⠳⢤⠔⢄⠀⠠⠹⢀⣀⣣⠞⢀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢘⠢⠏⠡⠡⠈⡀⠸⢹⡞⠀⢆
⡠⠃⠀⢀⣠⠛⠃⠀⠀⠢⠜⢸⠧⡄⢠⠸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢉⡁⣠⣘⢄⢀⢇⢀⣇⣰⡼⢻⡖⠊
⠃⠤⠊⠣⣀⡱⢦⡡⠌⣐⡰⠃⠑⠒⠂⠀⠀⠤⢀⠐⢌⣭⣔⡒⡒⢄⠀⠉⡧⠃⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⠥⠤⠤⠴⠄⢻
⠀⠀⠀⠀⠀⠀⣀⠤⠤⢄⡀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠠⠤⢀
⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠈⠀⠰⠀⠤⠤⠤⠤⠖⠊⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠤⠅⢁
⠀⠀⠀⢸⡄⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐⠊⠀⢄⠱⠄
⠀⠀⠀⠈⢌⠒⠄⢉⠒⠀⠐⢊⠅⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⣀⠄⠂⠠⢀⣀⠀⠈⢈⡀⠀⠀⢀⢠⢖⡀
⠀⠀⠀⠀⠈⠱⠤⣀⣀⡩⠝⠁⠀⠀⠀⠀⡠⣄⣀⠀⠀⠀⠢⠀⢈⣁⠀⠀⡀⠐⠈⡄⠀⠀⠈⠀⡑⠏⣠⠲⠊
⠀⠀⠀⠀⢀⠃⡐⠀⠀⠀⠀⠀⠀⣀⠠⠚⠒⠀⠘⣤⣄⠀⠀⠀⡀⠈⠐⡄⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⢀⠎⡘⠀⠀⠀⠀⠀⢴⡋⠄⠀⠀⠀⠀⡊⠀⠈⠉⠁⠚⡍⢣⠀⠷⠊⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠈⢄
⠀⠀⠀⢎⣠⠀⠀⠀⠀⠀⠀⠠⢈⠑⠈⠀⠙⠉⠈⠖⠉⠙⠆⠀⡠⠸⠁⠀⠀⢀⠔⠊⠁⠀⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠚⠀⠈⠓⠀⠉⠓⠈⠀⠀⠀⠀⢀⠏⠀⢀⠔⢋⠑⡦⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠑⣄⠀⠀⠀⠀⠀⠸⡀⠉⡇⠀⣭⠤⠣⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠲⢚⢉⠉⢫⡤⢀⠃⠀⠀⠀⠀⠀⠙⢤⡛⢮⣀⣀⡧⢏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠄⠂⠉⠀⠑⠄⠀⠈⣎⡤⠊⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⢂⠠⡀
⠀⠀⠀⠀⢲⠠⠠⠅⠀⢂⠑⠄⡄⣀⡠⠴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡘⢆
⠀⠀⠀⠇⠀⡑⢒⠔⠣⠤⠄⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠃

⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢠⠘⠃⠀⠀⠀⠀⠀⠀⡘⠆⠀⠀⠀⢀⣠⣼⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡂⠒⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠰⠶
⠀⠀⠱⡀⠀⠀⠃⠀⠀⠀⠀⠤⣙⡠⠀⠀⠀⠰⠵⢞⣯⠉⠃⠀⠀⠀⠀⠀⡇⠀⠀⠘⠃⣣⠀⠛⠁⠀⠀⠀⠀⠀⣾⣿⣿⡆⠀⠀⠀⠰⠆
⠀⠀⠀⢓⡀⠀⢰⠃⠀⠀⠀⠶⠆⣮⡄⠀⠀⠐⣄⠀⢹⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠈⠉⢠⣐⡾⠶⠀⠀⠀⢠⣿⣿⡿⣷⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠤⢀⣁⡋⠀⠀⠀⠸⡆⠘⠀⢠⠆⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠻⠏⠀⠀⠀⢄⠀⠾⠇⢸⠀⢿⠟⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⢰⠈⢅⣀⠀⠀⠀⢻⣄⢠⠃⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⢠⠄⡀⠀⠀⠀⠙⣆⠀⢘⡶⠀⢀⡞
⠀⠀⠀⠀⠠⣼⠎⠀⠀⠀⠪⡛⡚⠀⢙⠦⠀⠀⠀⠈⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠀⠀⠀⠀⠀⠀⠈⣇⢸⠃⠀⠘
⠀⠀⠘⢃⠔⡻⠁⠀⠀⠀⠀⠁⠍⠡⠂⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⣴⣾⣿⠒⠲⠀⠀⠀⠀⢹⡏
⠀⠀⠀⠈⠸⠅⠀⠀⠀⠀⠀⠀⠀⢻⠘⠋⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠘⠃⠀⠀⠀⠀⠀⢸⡇⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⡶⡀⠀⠸⣷⣿⠀⠀⠀⠀⠀⠲⠞⠂
⠀⠀⠀⣠⣤⣄⡀⠀⠀⠈⢲⣇⡂⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠠⠀⠀⠀⠀⠀⠰⣾⣶⡄⠀⠀⠀⠀⠸⣛⣿⠏⠀⠀⣿⡏⠀⠀⠀⠀⠐⠀⡀
⠀⠀⠼⠿⣿⣿⡇⠀⠀⠐⠀⠉⠵⠀⠀⠀⠠⣴⠤⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⠄⠛⡉⣶⡆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠈⢡
⠀⠃⠀⠀⠸⠻⠤⠀⠀⠀⣠⠁⠀⠀⠀⠀⠈⠀⠤⠀⠀⠀⠀⠉⡀⠉⠀⠀⠈⠀⠀⠀⡀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠘⢀
⠀⠂⢠⡶⣶⡶⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠑⠤⠤⠀⠀⠀⢶⣧⠄⠀⠀⠀⠀⠀⠀⠤
⠀⡀⠘⣾⠿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁
⠀⠅
⠀⠄
⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠓⣲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⠀⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢦⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡀⠀⠀⠉⠉⠛⢶⣄⡀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢻⡇
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢳⡀⠀⠀⠀⠀⠀⠉⠳⢌⠀⣠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠹⠃
⡼⠁⠀⠈⢆⠀⠀⣀⡀⠀⠀⠀⢶⠘⠻⣆⠀⠀⢀⡤⠂⠀⠀⠙⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠂⠀⠀⠀⠁
⢷⡀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⠓⠀⠁⠀⠀⠀⠀⠀⠀⠈⢄⣤⣀⣄⣦⣠⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠒
⠈⢷⡀⢦⠀⣈⣆⠀⠀⠀⠀⠀⢰⣲⡀⠀⠀⢠⣤⣄⠀⠀⢄⡀⠀⠀⢘⠟⠛⠛⠋⠉⠀⠀⠀⠀⠀⢀⣠⣤⡶⠏⠀⠀⠀⠀⠀⣄
⠀⠈⣷⣾⣇⠹⣿⣧⠀⠀⠀⢀⣿⢿⠆⠔⠛⠃⠊⠚⠻⣶⣿⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣴⡾⠛⢉⣡⣴⠀⠀⠀⣀⡤⠞⠃
⠀⠀⠘⣿⣿⣆⠹⣿⣆⠀⠒⠉⡁⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠉⠉⠉⠫⠶⡀⠀⠀⠀⠀⠀⢷⣠⣶⣾⣿⣿⣿⠶⠚⠋⠁
⠀⠀⠀⠘⣿⣿⣦⡟⡿⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠛⠛⠃⠀⠸⣍⠀⠀⠀⠠⣙⣦⣄⠀⠀⠀⢀⣀⡀⢠⠤⠖⠁⠀⠀⣠⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠈⠳⢦⣄⣠⣜⠻⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠹⡇⢸⣧⡀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣤⡀⠀⠀⠀⠀⢀⠠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠙⠻⣿⣶⣌⠙⠿⣿⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⢣⠀⠈⠻⢿⣷⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠹⢦⣄⣠⣀⡤⡤⢒⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠙⠾⣿⣷⣴⠿⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠁⠀⠀⠀⠉⠙⠢⣄
⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⡀
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠹⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⣀⡤⠴⠒⠋⠁⠀⠀⣀⡤⢾⡇
⠀⠀⠀⠀⠀⢸⠒⠚⠉⠉⠀⠀⠀⢀⣠⡤⠖⠋⠁⠀⢸⠁
⠀⠀⠀⠀⠀⠈⢦⣠⣤⠤⠔⠒⠋⠉⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠤⠴⢶⠛⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢝⢢⣀⡠⠴⢲⠶⠦⣄⡀⢧⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠠⡅⠉⠀⣰⠋⠀⣰⠃⠙⣎⣇⠹⣄⣀⣠⣄⣀⡀⠀⠀⣠⠴⠶⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⣰⠒⠀⠀⢀⡏⠀⢰⠇⠀⠀⠘⢻⡶⢫⠇⠀⣾⠀⠉⢲⠞⠔⠃⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢼⣘⣿⣠⠴⢦⠀⢸⠀⠀⣾⡋⠉⠭⣛⠿⠀⢸⡀⠀⢹⠀⠀⠀⠐⣃⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⡄⠀⡏⢳⡄⠀⠒⠀⠀⠀⣇⠀⠸⡆⠀⠀⢀⠀⠩⢤⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⡟⠀⡪⠀⣄⠀⠸⣄⠀⠹⡄⣠⠏⠙⠲⠯⠥⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠾⣢⠞⠉⠙⠲⠾⠶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠤⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⣡⠋⠀⠀⠀⢠⠎⠀⢉⠗⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠺⡉⠀⠳⡄⠀⠀⠀⠙⣌⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⡜⠁⠀⠀⠀⡴⠃⠀⡴⠃⠀⠀⠈⣣⡀⠀⠀⠀⠀⠀⠀⢀⣔⠁⠀⠀⠘⢦⠀⠈⢦⠀⠀⠀⠈⢣⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⣠⠏⠀⢀⡠⠤⠚⠒⠢⠾⣀⠀⠀⢀⠞⠁⢙⣄⠀⠀⠀⠀⣠⡋⠈⠳⡀⠀⠀⣀⠷⠐⠒⠓⠤⢄⡀⠀⠑⣄⠀⠙⣧
⠀⠀⠀⠀⠀⠀⡼⠁⢀⡴⠁⠀⡴⠋⠀⠀⠀⠀⠀⠀⠈⠳⣴⠋⠀⣠⠋⠘⡆⠀⠀⢰⠃⠙⣄⠀⠙⣤⠞⠁⠀⠀⠀⠀⠀⠀⠙⢦⠀⠈⢦⡀⠈⢧
⠀⠀⠀⠀⠀⢀⠃⣠⠎⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡜⠁⠀⠀⢹⡀⢀⡏⠀⠀⠈⢣⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠱⣄⠘⡀
⠀⠀⠀⠀⠀⢸⡔⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠂⠉⠒⠲⢄⣠⠞⢣⡜⠳⣤⠴⠒⠘⠙⠒⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢢⡇
⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⠞⠱⡀⠀⠀⠀⠀⠀⢠⠋⠀⣠⠖⠒⢲⠲⣄⠙⣆⡸⢦⠞⢁⣴⠒⡖⠒⢦⡀⠈⢧⠀⠀⠀⠀⠀⠀⢀⠏⠳⡀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢱⡀⡰⠋⠀⣰⠇⠀⠀⠀⠀⠀⢸⠀⠀⡃⠀⠀⠀⣟⠈⢆⠈⢧⠎⢠⡏⠀⡿⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠸⣄⠀⠙⢆⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠚⠉⠀⠀⠀⠀⠀⠀⠘⣄⠀⠳⣀⠀⠀⣘⣦⡻⠖⠋⠓⠾⣙⣶⡃⠀⢀⡰⠃⢀⡞⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡈⠉⠉⠀⢸⠁⠀⠀⠀⠀⢱⠀⠈⠉⠉⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢒⡶⠛⣄⠀⠀⠀⢀⠞⢦⡖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⡼⠋⢶⡘⣧⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⣠⡎⠀⠀⠀⡹⠁⠑⣄⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⣠⣞⠁⠙⢆⠀⢰⡇⠀⡰⠋⠱⣄⠀⠀⠀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⡉⠀⠀⢀⡤⢊⠏⠈⢣⡀⠈⢳⠇⠸⡞⠁⢀⡜⠁⢹⡦⡀⠀⠀⢈⣓⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡽⣁⡤⠚⠉⠀⣞⠀⠀⠀⠙⣄⡞⠀⠀⢳⣠⠋⠀⠀⠀⢳⠈⠓⠦⣌⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠁⠀⠀⠀⡰⠙⢦⠀⠀⠀⡜⠀⠀⠀⠀⢣⠀⠀⠀⡴⠋⢆⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠳⡄⠀⠳⡄⡰⠁⠀⠀⠀⠀⠈⢆⢠⠞⠀⢠⠞⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠘⢆⠀⢸⠃⠀⠀⠀⠀⠀⠀⠘⡇⠀⡰⠃⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⢳⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⡞⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠄⠀⠁⠐⣤⡀⠀⠐⡆⠀⢰⣧⠠⠤⠄⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⢰⣶⣶⠚⠀⠀⠁⠀⠸⣿⣆⢀⠂⠠⠈⢃⡀⠀⠀⠀⠀⠙⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⡀⢰⣿⣿⣀⠀⡠⠋⠀⡀⠀⣿⣿⠀⡀⠰⡇⠈⣅⠀⠀⠀⢦⣄⣤⣿⡿⠶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠢⣄⢘⣿⣿⣿⣿⡟⢹⠃⢀⡶⠀⠀⣿⣿⡄⠀⠊⣿⠀⠸⠀⠀⠀⣼⣿⣿⣯⠀⢠⣿⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⢰⣿⣿⣿⣿⣿⡿⠅⣸⠀⡆⢶⠀⠀⣿⣿⡌⠀⠀⣿⠀⢠⠠⡆⠸⣿⣿⣿⣷⣠⣿⡟⢐⡟⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡏⠈⠉⢟⠋⢻⣿⣿⣷⡾⠁⠘⡇⢸⡆⢸⣿⣟⡟⠃⠈⣷⠀⠼⢠⡇⠀⣿⣿⣿⣿⣿⣿⠃⣾⡇⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠓⠀⠀⠀⣠⠄⠸⠷⣌⡇⠀⠀⣁⣸⡄⣼⣿⣿⡇⠃⠀⢿⡀⠘⢺⠁⠀⡿⠟⠻⣿⣿⣿⣷⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣄⡴⠶⠞⠁⠲⢲⣀⠈⠳⣤⡀⠰⣾⡇⢹⣿⣯⣇⠀⠀⡿⠀⠀⠃⠀⡘⠁⠀⣹⣿⣁⣉⢛⣻⢿⣾⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠻⣦⣀⠑⠀⣯⣇⣼⣿⣿⣿⣥⣀⡇⠀⠀⡇⠠⠁⢀⡴⠋⠁⠀⠙⠟⠛⠛⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠉⢳⣀⡗⢿⣿⡟⢯⡟⢿⣽⣷⡀⠀⡇⢀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢀⡾⠧⣀⠀⠀⡠⣠⡄⠀⣸⣿⣇⣬⠅⠻⢿⣥⣿⢟⣯⢟⡀⣷⡏⠀⢠⠀⠀⠀⠰⠃⠀⠀⢀⠀⠀⠛⣷⣄
⠀⠀⠀⠀⠀⢰⠂⠀⠸⣁⠀⠋⠀⠀⠀⠐⠆⠀⠀⠀⢀⣾⣿⡏⠠⠀⣹⣲⣿⣿⡒⡳⣾⣿⣾⡇⠀⠘⣄⠀⠀⠀⠀⠀⠀⠈⠁⢄⡠⢾⣿⣇
⠀⠀⠀⠀⠀⠀⢠⠆⠀⡀⠀⠀⠀⡀⠀⠀⠀⣤⠀⠀⢺⣿⣿⠃⡠⢾⡟⠉⠉⣉⠛⠿⢄⠙⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡙⣧
⠀⠀⠀⠀⡀⠀⠀⠀⠔⠻⠀⠀⠈⠀⠀⠀⠀⠘⠀⠀⣿⣿⣿⠊⠀⠌⣷⡇⡟⠶⣶⡣⠄⠡⠈⡆⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣼⢿⣿
⠀⠀⠖⠀⠡⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⣀⠙⢶⣤⠐⢬⣅⡀⠀⠀⣸⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣟⡫⣧
⠀⢐⠃⠀⠀⠀⠁⠀⠀⠆⠀⠀⠐⠀⠀⠀⡆⠀⠀⣾⣿⣵⣿⡤⣿⣯⠀⠠⣿⠀⠀⣸⣿⣼⢶⣽⣿⡗⢄⠀⠀⠐⠀⣀⠀⠀⠀⡇⠀⠈⠈⢰⠹⡇
⠀⠒⠰⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⠁⠀⣸⠗⢿⡧⠘⢰⠏⠀⠀⠀⣿⠀⠒⢈⣿⡿⠀⠈⠛⠀⠀⠀⢀⡀⢀⣼⠂⠀⠘⠣⠀⠀⠀⣾⠁⠻⡄
⠀⠂⠀⡄⠀⠀⢀⠀⠀⠀⠀⢀⣀⣠⠴⠃⣠⠴⢡⣴⡿⠏⡤⠊⠀⠄⢠⠀⣿⠀⠀⠀⣄⡙⢄⣸⣧⣤⡗⢐⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⣠⡟⠆⠀⡿
⠀⡉⠀⠀⠀⠀⡆⠀⠀⣠⠖⠁⠀⠀⠀⠀⣴⠞⠋⠁⠐⠊⡁⠀⠀⠀⡿⠀⢛⠀⠀⢠⣿⡿⣾⣿⣿⣌⡛⠿⣄⢰⡀⠀⠀⠠⣶⠀⠀⢸⡿⢃⠀⣹⠇
⠀⠀⠀⠇⠀⠀⢹⡀⠀⠀⠀⠀⢀⣤⣾⠛⣉⣀⡁⣀⣰⣯⡊⣠⠴⢶⠎⡉⡛⠁⠑⠾⣗⠙⠃⣈⣷⡙⠣⠁⠋⠻⣷⣄⠀⠀⠀⢨⣘⠿⣯⣧⣽⣿
⠀⠀⠀⠈⢄⠀⣸⠃⠀⣠⣶⣿⡿⡟⡽⡟⡏⢹⢿⡿⣿⣍⠻⠇⣠⣤⡇⠀⠦⠀⢤⣀⣿⡷⠊⠁⢨⣿⣶⠈⢱⢴⢮⣍⣷⣄⠀⠀⠈⣻⣿⣿⣿⡿
⠀⠀⠀⠀⠘⡗⠁⢀⣴⢿⣿⡟⠁⠀⠠⡏⠇⠀⠀⠐⡘⡮⡁⢤⡩⣤⣼⠀⠁⠀⣌⣀⣿⡥⢒⣥⣾⠏⡞⢠⠀⠀⠀⡜⠁⠋⠻⣷⣾⣿⢟⣽⣿⠁
⠀⠀⠀⠀⠀⠹⣦⣾⣳⣿⣿⠁⠀⠆⠄⠈⠊⢄⠀⢶⣲⣑⣿⣆⠳⡉⠙⠀⡅⢐⢋⣿⡟⣰⣿⡿⠋⣠⠁⠎⡃⡄⡀⣅⠀⠀⡧⢹⣭⣽⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠄⠀⡴⣤⠀⠠⠘⣦⣀⣽⣿⣿⣿⣇⢥⠀⠀⢇⠨⣸⡿⣼⣟⣛⣥⣾⣯⠈⡰⡹⠃⢇⢅⠀⢀⡟⢸⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣍⣿⣿⡆⡄⢷⡘⢗⡀⠄⢀⠈⢻⣟⣿⣿⣿⣠⡄⠀⡀⢠⣿⣿⣿⣿⡿⠿⠟⢁⣸⡟⠁⢠⡾⡀⠀⣾⣧⣘⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⡌⠿⣟⢿⣿⣷⣦⣆⣠⢬⣙⠿⢿⣿⡇⢀⣲⠺⣿⣿⣿⣿⣿⠷⠛⣫⡏⣠⣠⠞⠁⡁⣰⣿⢳⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣟⡳⢮⣽⣿⣿⣿⣿⣿⣿⣾⣿⣷⣶⣴⣶⣿⡿⣿⣿⣿⣷⡾⠷⠾⠟⡉⣠⢠⣷⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣬⣍⣛⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡋⢁⠀⣠⢂⣔⣴⣧⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣝⡛⣿⣿⣿⣿⣿⣿⣿⠟⠘⣿⣿⠻⢿⣿⣿⣿⣾⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠿⠛⠉⠀⠀⠈⠛⠷⠿⠿⠿⠿⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠟⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣤⠀⠀⢠⠃⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠙⠀⠀⠀⠃⠀⠀⡆⠀⠀⠀⠀⠴⠆⢀⡈⠁⡀
⠀⠀⠀⠐⠃⢠⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠶⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⣀⠀⠙⠋⠀⠁
⠀⠀⠀⠀⣰⠘⠃⠀⠀⢀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⡀⠀⠀⠀⠟⠁⠀⡄
⠀⠀⠀⠀⠪⠈⠀⠀⠀⠈⠀⠀⠀⠘⠇⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠃⠀⠀⠀⠀⢠⢀⡄
⠀⠀⠀⠈⢠⡤⠄⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠀⠀⣈⠂
⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠹⠀⠀⠀⠀⠙⡄⠀⠆⡔⡡⠐⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢨⠆⠂⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⡆⣾⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣦⣮⠤⠄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠘⡿⠀⠠⠀⠀⡐⠀⠀⠀⢀⡨⠖⣯⣱⠘⠆⠀⠀⠀⣠⣶⡄
⠀⠀⠀⠀⠘⠰⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠊⠸⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠆⢰⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⠀⠀⠀⠐⡄
⠀⠀⠈⠀⠀⠠⠀⠀⢀⠠⠁⢠⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠀⠀⠀⠀⠱⣦
⠀⠀⠀⠑⠆⠀⠀⠀⢀⠤⠘⢶⡷⠃⠀⠀⠀⣾⡻⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣀⠠⡄⠀⠀⠀⡔⠁⠀⠀⠀⠀⠘
⠀⡂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠉⠑⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠃⠁⠀⠀⠀⠁
⠀⡀
⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣰⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠹⢦⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠶⠶⠶⣿⠀⠃⣰⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡠⠖⠋⣁⡤⠴⠒⡦⠀⢙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠒⢋⣡⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠻⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢲⡶⠶⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢰⡄⠀⠀⠀⠀⠙⠳⣄⡀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠈⠛⠶⣄⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣴⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠚⠛⠛⠃⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⣰⠟⠁⢀⠴⠚⠉⢉⡤⠆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⣶⡶⠋
⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⢠⣤⣀⣀⣠⣤⠴⠞⠉
⠀⠀⠀⠀⠀⠀⠈⠻⢦⣤⣀⣀⡀⠀⠀⢠⠇⢠⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣦⠀⠀⠀⡏⢀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠐⣇⡼⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠈⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠞⠁


⣿⣿⣿⣿⣛⣿⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣻⣟⣻⣿⣟⣛⣛⣿⣟⣻

⠀⠀⠀⣠⠞⣉⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣉⡙⢦⡀
⠀⠀⠀⣿⢸⠁⢹⠀⡇⡒⠒⠀⠀⠀⠀⠀⢸⠁⡏⠈⡇⢹⠠⠤⠄⠀⠀⠀⠀⠀⡇⢸⠁⣿⢸⡇⡒⠒
⠀⠀⠀⣿⢸⠀⢸⠀⡇⠂⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⡇⢸⢨⠀⠀⠀⠀⠀⠀⠀⡇⢸⣄⣿⢸⡇⠂
⠀⠀⠀⣿⢸⠀⢸⠀⡇⠥⠤⠀⠀⠀⠀⠀⢸⠀⡇⠀⡇⢸⢐⣀⡀⠀⠀⠀⠀⠀⡇⢸⣿⣿⢨⡇⠥⠤
⠀⠀⠀⡟⢸⠀⢸⠀⡇⡁⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⡇⢸⠐⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⢰⡇⡁
⠀⠀⠀⣷⢸⠀⢸⠀⡇⣂⣀⠀⠀⠀⠀⠀⢸⠀⡇⠀⡇⢸⠨⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⢰⡇⣂⣀
⠀⠀⠀⣿⢸⠀⢸⠀⡇⠆⠀⠀⠀⠀⠀⠀⢸⠀⣷⣾⡇⢸⢈⠉⠁⠀⠀⠀⠀⠀⡇⢸⣿⣿⢸⡇⠆
⠀⠀⠀⣿⢸⠀⢸⠀⡇⠅⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⡇⢸⢐⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⢨⡇⠅
⠀⠀⠀⣿⢸⠀⢸⠀⡇⡋⠉⠀⠀⠀⠀⠀⢸⠀⣿⣿⡇⢸⠰⠒⠂⠀⠀⠀⠀⠀⡇⢸⣿⣿⢸⡇⡋⠉
⠀⠀⠀⣿⢸⠀⢸⠀⡇⡂⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⡇⢸⠨⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⢸⡇⡂
⠀⠀⠀⣿⢸⣷⣿⠀⡇⠍⠉⠀⠀⠀⠀⠀⢸⠀⣿⣿⡇⢸⢘⠒⠂⠀⠀⠀⠀⠀⡇⢸⣿⣿⢸⡇⠍⠉
⠀⠀⠀⣿⢸⣿⣿⠀⡇⠅⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⡇⢸⢐⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⢸⡇⠅
⠀⠀⠀⡏⢸⣿⣿⠀⡇⠒⠒⠀⠀⠀⠀⠀⢸⠀⣿⣿⡇⢸⠨⠤⠄⠀⠀⠀⠀⠀⡇⢸⣿⣿⠐⡇⠒⠒
⠀⠀⡴⢃⣼⣿⣿⣄⠳⡄⠀⠀⠀⠀⠀⢠⠞⣠⣿⣿⣧⡘⢦⠀⠀⠀⠀⠀⠀⡴⢃⣼⣿⣿⣄⠳⡄
⠀⢸⡇⣾⣿⣿⣿⣿⡆⢻⠀⠀⠀⠀⠀⡟⢰⣿⣿⣿⣿⣷⢘⡇⠀⠀⠀⠀⢸⠃⣿⣿⣿⣿⣿⡆⣿
⠀⠈⢧⠙⢿⣿⣿⠟⢡⠟⠀⠀⠀⠀⠀⠹⡌⠻⣿⣿⡿⠋⡼⠁⠀⠀⠀⠀⠈⢧⠙⢿⣿⣿⠟⢡⠏
⠀⠀⠀⠙⠶⠤⠴⠖⠉⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⠤⠶⠋⠁⠀⠀⠀⠀⠀⠀⠈⠑⠶⠤⠴⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡸⠀⠀⠀⠀⠀⠀⠉⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⠻⡍⠉⢻⣧⠀⠀⠀⠀⢀⣀⡤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠉⠳⡙⢆⠀⠹⣷⡤⠖⠚⠉⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⡤⣄⠀⠀⠙⣌⠳⣄⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⡽⠯⢍⠙⢎⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠓⠁⠀⠀⠉⠁⠀⠀⠀⠈⠳⢌⡵⠁⠀⠀⠀⠀⠀⠀⠀⢰⠀⡏⢜⢍⡡⡡⢈⠀⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⡏⠀⠾⠇⠀⠀⣀⡀⠀⠀⠀⠮⠆⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⣓⣒⡡⡾⣺⣭⣭⣍⡛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠘⢞⢦⠀⠀⠀⠀⠓⠃⠀⢀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠸⢸⠑⠕⠢⠑⠹⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠴⢻⣆⠈⢣⡳⡀⠀⣀⠀⠀⠀⡸⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢠⠘⣜⢪⣖⡰⢠⢀⠟⡇
⠀⠀⠀⠀⠀⠀⢰⡉⠀⠀⠀⠀⠀⠹⣧⡀⠑⢌⠢⣟⣡⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠓⠤⢭⣥⡴⠟⠁⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⠀⠀⠀⠀⠈⣻⢦⡤⠟⠋⠀⠀⠀⢀⡠⠤⠀⠀⠀⢀⠴⠊⠁⠤⢦⣤⣍⠢⡀⠀⡇⠉⠉⠙⠒⠒⠉⠉⠁⢀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠎⢂⠘⡀⠈⠉⠒⠒⠒⠒⠒⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢀⡠⠋⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⡤⠴⢒⣒⣲⣶⣤⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡱⠁⡔⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠂
⠀⠀⠀⠀⠀⠀⢀⣴⡂⢭⠕⠒⠈⠉⢋⠀⠀⠀⠉⣷⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⢀⡜⠀⠀⠀⠀⢀⣠⠔⠋⠁⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠏⠁⠀⠀⠇⠀⠀⠀⣈⣤⠤⠄⠂⠺⡇⠀⠀⠀⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⡰⠋⠉⠓⠲⢤⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠸⣀⣠⠤⠾⠖⠊⠉⠁⠰⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠈⠓⢤⠁
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢃⠀⣀⣀⡠⣧⠒⠒⠈⣫⡆⠀⠀⠀⠀⠀⠀⠈⠳⣤⡴⠋⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠸⡤⠖⠚⠛⡏⡡⠢⡄⣸⠆⠉⠉⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠
⠀⠀⠀⠀⠀⠀⠀⡠⠛⠒⠴⠒⠛⠁⠀⠈⣀⢀⡠⠤⠤⣤⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢀⡎
⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣀⡤⠐⠊⠁⠈⠁⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠇⠀⠀⠀⠀⢀⡀⠀⢠⡄⠆⠢⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡤⠴⠚⠈⠂⠉⠈⠀⠀⠀⠀⠀⢋⡆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠲⢄⡀⠀⠀⠀⠀⢀⠊⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⠀⠈⠓⠢⢄⡠⠃⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡇⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⡆⠀⠀⣀⡤⠖⠋⣇⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⠀⠀⠀⠀⠀⠀⡇⠇⠀⠀⠀⢁⠴⠊⠁⠉⠁⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠤⠤⠤⠤⠔⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢇⡅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⡄⠐⣢⣤⡀⠀⠀⠀⠀⠀⢀⣴⣆⠂⣤⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⡿⠛⢿⣿⣿⣿⣿⣿⡿⠛⢻⣿⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠻⡷⠊⠛⡋⠛⡀⠀⠀⠀⠀⢀⣬⠉⠙⠋⡁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣦⣈⣿⣤⢴⣮⣿⣿⣿⣿⣿⡿⢷⣾⣧⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣦⣰⠀⠀⣿⣦⡇⠀⠀⠀⢰⠟⡃⠀⠘⠾⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⠻⢿⣿⣿⡄⠙⢻⣿⣿⣿⣟⣠⣼⣿⣯⣡⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢠⠙⢧⠀⠀⠀⣤⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⡿⣦⣸⣿⣿⣿⠟⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣮⠀⠀⠠⢋⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⠛⣿⣿⣿⡶⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠻⡄⠀⣼⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣦⣿⣿⣇⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡇⢀⣤⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠙⢻⣿⠛⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡈⢳⠸⣋⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢷⣼⣧⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠻⣿⡇⠀⠀⠀⠀⣾⣟⣼⠄⠀⠀⠀⠀⢠⣴⣦⡄⠀⠀⠀⠀⣠⡐⡦⣀⠀⠀⠀⠀⠀⠀⠂⠐⢦
⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⢀⣾⣿⣿⣿⡄⠀⠀⠘⣷⢣⢠⡍⠀⠀⠀⠀⡀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠙⠒⠛⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣴⡄⠀⠀⠀⠀⡰⣟⡲⠀⠀⠀⠀⠀⢠⣶⣶⣄⠀⠀⠀⠀⠀⣀⢸⡶⠀⠀⠀⠀⠀⢠⣾⣷⣄
⠀⠀⠀⠀⠀⢠⣿⣿⡗⠀⠀⠀⠀⠀⠿⠮⠙⠁⠀⠀⠀⢀⣿⣿⣿⣿⡆⠀⠀⠀⠰⣬⡳⠁⠀⠀⠀⠀⠀⠘⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⡿⠃⠀⠀⠠⣿⠊⠁

⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠈⠉⢙⡇⠀⠀⠀⠀⠀⣾⣿⣿⣷⠀⠀⠀⠀⠀⢠⣴⣿⠃⠀⠀⠀⠀⢠⣿⣷⡆⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠻⢿⡿⠏⠀⠀⠀⠀⣸⣷⡿⠋⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠾⣟⡀⠀⠀⠀⠀⣴⢿⣿⣷⡄⠀⠀⠀⠀⣀⣾⣿⣦⠀⠀⠀⠀⢾⣿⣿⠇⠀⠀⠀⠀⣶⣤⣴⣦⣶
⠀⠀⠀⠀⠀⠠⣤⣤⠀⠀⠀⠀⠘⠁⣾⣿⣿⡿⠀⠀⠀⠀⠉⠛⣛⣽⠃⠀⠀⠀⠀⢹⠃⠀⠀⠀⠀⠀⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⣀⣴⡄⠀⠀⠀⠀⠀⢠⡴⠀⠀⠀⠀⠀⢀⠄⢼⠷⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠆
⠀⠀⠀⠀⠀⢠⣾⣿⡿⠀⠀⠀⠀⢀⠮⠁⠀⠀⠀⠀⠀⠐⡃⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦⡀⠀⠀⠀⠀⢀⣯⡀
⠀⠀⠀⠀⠠⣾⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠇⠂⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠃⠀⠀⠀⢰⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⢛⢿⠛⡟⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠉⠉⠉⠉⠉⢉⣭⣭⠉⠉⠉⠉⠉⠉⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠉⢹⣿⣿⠉⠉⠉⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⡀⠀⠀⠑⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠛⣿⣀⣩⡿⠉⣷⡀⠀⠀⠀⠀⠀⠀⢀⣴⠋⣿⠋⣴⡈⠉⠛⣦
⠀⠀⠀⠀⠀⠀⠀⣼⠋⢱⣾⡿⠛⠛⢷⡶⠿⢷⡄⠀⠀⠀⠀⣠⣿⣧⢸⡿⠟⠛⠻⣟⠛⢿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⡷⣄⣿⠀⠀⠀⠀⢿⣶⣤⣿⡀⠀⠀⢠⣿⠐⣿⠏⠀⠀⠀⠀⣿⣇⢠⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣏⠻⠿⠀⠀⠀⠀⠈⣿⣉⡙⣷⠀⢀⡾⢏⣶⡏⠀⠀⠀⠀⠀⢏⠈⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⢄⡀⠘⣟⠻⠿⢧⣼⠇⠈⡟⠀⢀⡠⠤⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠛⠶⢤⣿⠻⣾⡿⠁⠀⠁⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠕⠃⠉⣙⣻⣶⣿⣼⣷⣃⡉⠋⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⡶⠶⠛⠟⠉⣹⠛⢿⣏⠉⠉⠛⠳⠶⣦⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⣸⠿⣠⣾⠉⢷⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡄⢠⢂⣿⡀⣹⡟⠻⣿⡆⠹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠻⠇⣾⠻⣿⡟⢿⣦⣬⣿⡀⠹⠣⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣦⣼⠃⠘⣿⠛⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠛⣿⠏⠀⠀⠹⣷⣠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡧⣤⡟⠀⠀⠀⠀⢻⡍⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣈⡿⠁⠀⠀⠀⠀⠘⣿⠯⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠉⠓⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡟⠻⠯⠤⠤⠤⠤⠭⠭⠭⠽⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣤⣭⣙⣒⣒⣒⣒⣒⣒⣒⣒⣚⣩⣬⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⠁⡏⢹⠉⡏⢹⠉⡏⢹⠉⡇⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣷⠾⣦⣧⣼⣤⣧⣼⣤⣧⣼⣤⠷⣾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠻⠶⠶⠶⠶⠶⠶⠶⠶⠟⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠛⠿⠿⠿⠿⠿⠿⠛⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠐⠶⠶⢶⡶⠶⠶⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠐⠻⠶⠶⠶⠂⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⠀⠀⠀⠀⢀⡴⡖⢶⡄
⠀⠀⠀⠀⠀⢀⡴⠶⣤⣤⣄⡀⠀⠀⣿⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁⣿⡀⠀⠀⣴⠟⢮⣁⣼⡇
⠀⠀⠀⠀⠀⢸⣄⣀⣸⠐⠚⣻⠀⠀⣿⣦⣄⣀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣿⣿⠆⠘⣧⣀⣴⠿⠋
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠿⠿⠋⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⢶⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣉⣸⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠙⢳⣀⣄⠀⢀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⢠⠁⠈⠉⢉⠉⠊⢁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⠐⠀⢀⡄⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣼⣦⣀⡞⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠐⠛⠻⠿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⠂⠀⠀⠀⠀⠽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⢨⠆⣄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠑⠄⠀⠐⠙⣿⣷⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠈⠢⣰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠠⠀⠀⠀⠀⠈⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⢀⡀⣀⠀⠀⠀⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⠢⡀⠀⠻⣻⣷⡄⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⠈⠢⣀⢽⣿⣿⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡦⡀⠀⠀⠈⢺⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠈⢀⠀⠀⠀⠹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣾⣦⡐⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣩⠟⣿⡟⣟⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠐⠁⡀⠸⠃⡯⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣠⠊⣹⠊⠀⠀⣀⡳⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⠴⠁⠙⠒⠒⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣤⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀
⠀⠀⠀⠀⢽⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠉⠒⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⡀⢄⠀⠔⠀⠀⠀⠀⠀⠀⢀⣼⣿⡣
⠀⠀⠀⢱⣿⣿⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠠⢬⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠘⠂⡁⠀⢁⠄⠀⠀⠀⢤⣿⣿⣟⣅⠡⠂
⠀⠀⠠⣕⣲⣿⣿⣭⡁⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠈⠀⠀⠀⠀⠀⠩⡷⠀⠀⠀⠀⠠⠬⣀⠾⠀⠀⠀⠀⢀⣹⣯⣿⣿⣩⣶⠥⠄
⠀⠀⠨⣓⣿⣿⣷⣖⡂⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⣈⠡⠔⠀⠀⠀⠲⣿⠀⠀⠀⠀⠀⠁⠂⢀⠀⠀⠀⠀⢉⣹⣿⡿⣯⣤⠵⠖⢒⣀⡀
⠀⠀⠀⠤⣿⣿⣓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⠅⠀⠂⠀⠀⠀⠀⠁⠂⠀⠈⣙⣓⠦⣽⣿⣡⠴⠖⢛⣉⣁⡀
⠀⠀⠈⠽⣿⡲⠬⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠨⢇⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢿⣶⡶⡼⣻⣯⣭⠥⠖
⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⢤⣈⣹⣿⣫⡶⠶⠖⢚⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡧⠄⠀⠀⠀⠀⠀⠀⢠⡃⠀⠘⠻⢷⡴⠀⠀⠀⠀⢀⣉⣉⣁⠾⣛⣋⣉⡀
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⢠⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠘⣉⣉⣀
⠀⠄⠀⠀⠁⠀⢰⡄⠀⠀⠀⣇⠀⠀⠀⠀⠀⣰⣇⢀⠀⠀⠀⠀⡌⠀⠀⠀⢀⠀⠀⠁⠀⠰⣄⠀⠀⠀⢠⡆⠀⢠⣴⣤⠀⠀⠁⠈
⠀⠀⠀⠀⠀⡐⣾⣯⠀⠀⠀⣿⠄⠀⠀⠀⠀⢨⢉⡁⠀⠀⠀⠀⣇⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡄⠀⠀⢸⠀⠀⠘⢻⡍⠀⠀⠀⠀⣤
⠀⠃⠀⠀⠀⢸⣿⣷⠀⠀⠀⡿⠀⠀⠀⠀⠀⠠⠀⢀⠲⠃⠀⠀⣿⠀⠀⠀⠘⣿⠀⠀⠀⠀⢸⠇⠀⠀⠈⠀⠀⠀⠀⢻⠀⠀⠀⠀⣿
⠀⡊⢸⠀⠀⠈⢸⡇⠀⠀⠀⡇⠀⠀⣾⡀⠀⠀⠀⠀⠁⠀⠀⠰⠿⢤⠄⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣪⠀⠀⠀⠀⠟
⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠂

⠀⠀⠀⣀
⠀⠠⣾⣿⣿⠀⠀⠀⠀⠀⠀⢰⣾⣤⠀⠀⠀⠀⣄⣇⣇⡄⠀⠀⠀⠀⠠⡄⠀⣠⡖⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⣦⠀⣠⡇
⠀⠀⢻⣿⣿⣠⠆⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠈⠙⣿⣿⠆⠀⠀⠀⠀⠸⣿⣸⣯⠄⠀⠀⠀⠐⣿⠁⠀⠀⠀⠀⠳⣾⣷⣿⣷⠆
⠀⢀⣸⣿⣿⠋⠀⠀⠀⠀⢦⣸⡇⠀⠀⠀⠀⠀⣈⣟⡟⠀⠀⠀⠀⢳⡔⣿⠈⠁⠀⠀⠀⠀⢠⣹⠀⠀⠀⠀⠀⠠⠟⢻⠛⠉
⠀⠀⢻⣿⣷⠀⠀⠀⠀⠀⠻⢿⡇⠀⠀⠀⠀⠀⠙⣿⣇⠀⠀⠀⠀⠈⠻⡿⠀⠀⠀⠀⠀⠀⠀⢹⣿⠆⠀⠀⠀⠀⠀⣿
⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⢀⢸⣷⡆⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣶⣹⡀⠀⠀⠀⢲⡀⣿⠀⢀⡀⠀⠀⠀⣿⡄⣰⡇
⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⢼⡇⠀⠀⠀⠀⠀⠀⣿⠀⣠⡀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠈⢷⣿⢠⣿⠁⠀⠀⠀⣿⣿⡿
⠀⠀⢸⣿⣿⣯⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣳⣄⣿⣾⣥⣌⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⣿⣾⣷⠃⠀⠀⠀⣿⡟⠁
⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⠉⠈⣿⠉⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⠀⣿⡇
⠀⠀⠘⠛⠛⠛⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠰⣆⢠⡇⢀⡄⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠲⣄⣿⣦⡆⠀⠀⠀⠀⢠⠀⣤⢠⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢿⣿⣷⡿⠀⠀⠀⠈⣿⡄⣰⡟⢧⠀⠀⠀⠉⣻⠋⠀⠀⠀⠀⠀⢸⣤⣿⣾⠀⠀⠀⠀⣨⠜⠿⠒⠀⠀⠀⠀⢠⢿⣧⡀
⠀⠀⠻⡿⣿⠗⠀⠀⠀⢠⡾⣷⡟⠀⠀⠀⠀⠀⢈⡹⡰⠆⠀⠀⠀⠰⣼⣿⣿⣿⠖⠀⠀⠈⠀⠀⠀⢧⠀⠀⠀⠀⡈⣼⠙⠈
⠀⠀⠀⢰⡆⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⢀⣾
⠀⠀⠀⢸⡇⠀⠀⠀⠀⢰⣾⣿⣿⣶⡆⠀⠀⠀⢿⡇⢠⣾⠃⠀⠀⠀⠀⣴⣦⡀⠀⠀⠀⠀⠀⣠⣶⠀⠀⠀⠀⠀⣠⣤⣼⡏
⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⣿⣾⣿⡧⠀⠀⠀⠘⣿⣿⠋⠀⠀⠀⢤⣾⣿⣿⣿⣤⠀⠀⠀⣾⣿⠟⠀⠀⠀⢠⣾⠟⣻⡇
⠀⠀⠀⠸⠟⠀⠀⠀⠀⠈⠉⠛⠛⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠙⠛⠿⠿⠛⠃⠒⠀⠐⠿⠛⠀⠀⠀⠀⠈⠀⠀⠻⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠴⠶⠶⡒⠛⠛⠛⢦⣠⠞⠁⠀⠘⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠛⢏⡉⠉⠀⠀⠈⠒⣄⠀⠈⢢⠀⠀⠈⣧⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⣲⠶⠤⣄⣀⠀⣼⠉⠀⠀⠉⠢⡀⠀⠙⢄⠀⠀⠀⠀⠈⢆⠀⠀⢣⠀⠀⢸⡅⠀⠀⠀⠀⢻⡀
⠀⠀⢰⡏⠀⠀⠀⠈⠙⣿⠀⠀⠀⠀⠀⠘⡄⠀⠈⡆⠀⠀⠀⠀⠈⡆⠀⠈⡄⠀⠈⡗⠶⠦⠤⠴⠶⠛
⠀⠀⠀⣹⠂⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⢰⠀⠀⢰⠀⠀⠀⠀⠀⢥⣀⣀⣧⡴⠾⠁
⠀⠀⠀⢧⡀⠀⠀⠀⠀⣠⢷⡀⠀⠀⠀⠀⢀⣆⣠⣼⡤⠴⠶⠚⠒
⠀⠀⠀⠀⠉⣳⢀⣤⠞⠁⠈⠛⠒⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠚⠋⠉⠛⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠈⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣼⠧⠤⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡾⠁⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⡞⠁
⠀⢀⣤⠶⠶⠶⠶⠶⣤⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟
⢠⡟⠉⠢⡀⠈⠠⡀⠀⠈⠙⠶⣄⣿⠛⠒⠒⠒⠓⠛⠛⠋⠉⠉⠉⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⣿⢧⠀⠀⠈⠳⣄⠈⢦⡀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⣿⠘⣆⠀⠀⠀⠈⠳⣄⠙⢦⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠸⣆⠈⢢⡀⠀⠄⣀⠈⠳⢄⠙⢦⣘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠂⠁⠀⢀⡾⠁
⠀⠘⢦⡀⠙⢦⡀⠀⠀⠀⠀⠑⢄⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣸⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠙⢦⣀⠈⠓⠶⢤⣤⣀⣤⢽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠈⠛⠲⠦⣤⣤⣤⠶⠟⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢀⣠⡴⠶⠚⠛⠛⠉⢉⣽⠟⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⣠⠾⠋⠁⠠⠄⠀⢀⡤⠞⠉⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⠀⢀⣀⣠⠴⠛⠁⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⡠⠚⠁⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠶⠦⠤⢤⣄⣀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣸⡃⠀⠀⣠⠖⠁⠀⣠⠖⠉⠀⠐⠀⠀⣠⠎
⠀⠀⠀⠀⣠⣴⡛⠛⠲⢤⣀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡷⢴⣟⡁⢀⣠⠞⠁⠀⣀⣀⡤⠖⠋⢁⡼
⠀⠀⠀⣰⠏⠈⠙⠲⢦⣀⡈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⠈⠉⠉⠉⠉⠉⢉⣀⣠⡤⠶⠋
⠀⠀⠀⣿⣄⠀⠀⠀⠀⠈⠙⠲⢤⣀⠈⠙⠲⢤⣀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⢹⣏⠑⠦⣄⡀⠀⠂⠀⠀⠈⠙⠲⢤⣀⠈⠙⠲⢦⣤⢿⡆
⠀⠀⠀⠀⠻⣆⠀⠀⠉⠓⠒⠤⠤⣀⣀⣀⣀⣬⠽⠶⠒⠉⠀⢀⡿
⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⣀⣀⠀⠀⣀⣀⣀⣤⡴⠞⠋
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⡀⠀⠀⣀⣸⣿⣀⡀⠀⠀⣠⣾⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠛⠿⣿⣦⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣿⠿⠛⠛⠻⢿⣷⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⣴⣾⣿⣦⠘⣿⣇⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⠈⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⠀⣿⣿⠀⢠⣤⣤⣤⣤⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⠀⣿⣿⠀⠈⠉⠉⠉⠙⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠹⣿⣧⡀⠀⠀⠀⠀⠀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⣠⣾⡿⠻⢿⣷⣶⣶⣶⣿⠿⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⠀⣿⣿⠀⠀⠀⠀⠸⠿⠋⠀⠀⠀⠀⢹⣿⠁⠀⠀⠀⠈⠻⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠀⣿⣿⣿⣿⠀⠻⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠁⠀⢀⣿⣿⣿⣿⡀⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⣠⣿⣿⣿⣿⣿⣿⣆⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠰⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠹⣿⣿⣿⣿⣿⣿⠟⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠈⠙⠛⠛⠛⠁⠀⢀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣷⣤⣄⣀⣀⣀⣤⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⣀⡤⢶⠚⠛⠛⠛⠛⠷⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠞⠛⠛⠛⠛⢷⢶⣤⣀
⠀⠀⠀⠴⠯⠥⢀⡀⠑⢄⠀⠀⠀⠸⠀⠈⡟⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⡇⠀⠘⠀⠀⠀⢀⠞⠁⣀⠠⠽⠷⣆
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⢂⠀⠀⢀⠀⠀⠃⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡎⠀⠀⠇⠀⠀⠀⠀⢀⠃⡠⠊⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⢀⣀⣀⡀⢀⡵⠾⢦⣤⣜⠀⡘⠀⠀⢀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢁⠀⠀⠘⡀⢘⣠⡤⠾⢶⣅⡀⣀⣀⣀⠀⠀⠈⢷
⣆⠤⠂⠉⠀⠀⢀⣠⠋⠀⠀⠀⠀⠙⠻⣄⠀⠀⡈⠀⡈⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⡀⠀⠀⠑⠋⠀⠀⠀⠀⠈⢿⡄⠀⠀⠈⠁⠢⢼
⡇⠀⢀⠠⠂⠉⠀⣇⠀⠀⠀⠀⠀⠀⠀⠙⣧⡐⠁⢀⠃⠀⠘⣷⡀⠀⠀⠀⠀⢠⠄⠀⠀⢇⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠈⠑⠢⢀
⣧⡰⠁⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠠⠞⣛⠛⠻⢿⣄⡀⠀⡏⢷⡄⠀⠀⡰⠁⡆⠀⠀⢈⠶⠖⠛⠻⢶⣄⠀⠀⠀⠀⠀⡰⡀⠀⠀⠀⠀⠡⣀
⢹⡄⠀⠀⠀⠀⠎⠀⡇⠀⠀⠀⠀⠀⠀⢰⡿⣶⣦⡀⠙⢻⣶⠁⢀⣿⣦⣄⣅⣀⣡⠀⠀⠁⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⡇⠈⠆⠀⠀⠀⠀⠘
⠈⢷⣄⠀⠀⠘⢀⡴⠁⠀⠀⠀⢰⠀⠀⠀⠙⠿⢿⣿⣴⣤⣙⠟⠉⠉⠉⠉⠉⠉⠻⣇⣠⠞⠁⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢷⡀⠘⠀⠀⠀⡀
⠀⠀⠙⠛⠒⠛⠋⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢻⢁⡠⠄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⠛⠳⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠒⠚⣻⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡠⠄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠚⣻⡄⠀⠀⠀⠀⠀⠀⠀⢸⡏⢀⣀⣀⣀⣀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠐⠊⡩⢿⣤⣤⣤⣤⣤⣤⡤⢾⣿⣭⣋⠉⠭⠩⢿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⢀⣀⣤⡴⠋⢻⣆⠈⠁⠀⠀⠀⠀⢻⣿⣝⠛⠻⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣙⡛⠛⠛⠙⡍⠀⡱⠀⠈⠸⢷⡀⠀⠀⠀⠀⠀⠙⠿⣦⣤⣤⣤⣤⠴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣏⠛⠻⠷⠶⠞⠋⠁⠀⠀⠀⠘⠛⣶⡄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⣴⣯⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⣉⣻⠟⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠙⣿⡻⢶⣤⣤⣤⣤⠴⣖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠐⠻⣿⣧⣤⣤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠈⢷⡊⠀⠀⠔⠁⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢃⠀⠀⠈⠂⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣔⠊⠀⠀⢀⠃⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⡠⠂⠀⡠⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠑⣄⠀⢈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⡀⢀⠔⠁⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⣀⣈⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣯⣀⣀⣠⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠀⠀⠈⠛⠳⠛⠛⠛⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣤⣤⣿⣄⡀⠀⠀⠀⢀⣤⡶⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⣿⣁⡽⢀⣀⠈⠛⣦⡴⠞⠋⠁⠀⢀⣠⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⢀⣀⠀⠀⠠⡏⣹⠇⢀⡼⣧⣠⣤⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣧⠀⣏⣈⡷⠀⠀⠉⠁⠀⠈⠓⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠶⢋⠀⠀⠉⡴⢤⡀⠠⡞⢳⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠰⣏⣹⠀⠀⠙⠦⠇⠀⠉⠉⣀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⡀⠈⠁⠀⠀⠀⣀⡤⣄⠀⢰⣣⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢿⡤⠇⠀⣞⣹⠆⠐⢧⡤⠏⠀⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉⠀⣸⣷⡀⠀⠀⢁⣤⣄⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉⠀⠀⣠⡾⠋⠀⠙⠷⢦⣼⡧⣾⠟⠋⠁
⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠾⠋⠀⠀⠀⠀⠀⢸⡇⠀⢹⡄
⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠶⠶⢦⣄⠀⠀⠀⠀⠀⠸⠃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⢀⣀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡴⠛⠉⠻⣆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋
⣿⣿⣿⣿⣛⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣛⣿⣿⣛⣟⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢠⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠐⠘⡇⡁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣷⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⡀⣸⣿⡜⠰⣆⠀⠀⠀⠀⠄⠀⠀⠀⢠⣯⠙⣩⡃⢶⣴⡄⣿
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣠⣤⢤⢰⡶⠔⢤⡝⠛⠉⠀⣆⠈⠀⠀⠀⢀⢠⡄⠀⠂⣿⠷⠿⢈⢥⢸⡿⠵⣏
⠀⠀⠀⠀⠀⠐⠛⢦⡀⡀⠿⠛⠀⢨⣧⠈⣦⣐⡇⠀⡘⠋⠀⠀⠀⠠⡾⢘⠁⠀⢠⣾⠓⠲⠆⣿⠿⠆⠀⣿
⠀⠀⠀⠀⠀⠀⢰⠎⣣⡁⠐⢀⡀⠆⠉⠐⠀⡌⡗⠀⠃⠀⠀⡀⡄⠀⠀⠀⢀⣀⠀⠉⢨⡄⣸⣗⣶⠦⠀⣿
⡀⠀⠀⣀⠀⣠⠀⣹⠋⠁⢻⣿⣏⠁⠀⠀⠀⠁⠃⠀⠀⠀⠄⢁⣗⣰⣄⣠⣤⣼⡄⠘⠋⣠⢁⣙⣿⣿⠆⣿
⠁⠀⠝⢻⡇⠁⢠⣟⣶⡆⣸⣿⠉⠁⢸⠀⢳⡀⠀⠀⠀⢰⠀⢸⣿⣽⣿⣿⣏⠈⡗⣶⣰⠃⣘⠛⣿⣿⡿⣿
⢀⡀⠀⠀⢁⣴⠏⡜⠀⡿⢿⠛⠀⠢⣽⠀⠀⢷⢀⣄⠀⠐⡀⢉⣽⣿⣿⣹⣿⣷⡀⢠⣷⡔⠟⠀⢻⡉⠃⣿
⠃⢐⣃⠀⡸⢿⣀⡁⠀⣈⣭⣄⣠⣶⣿⠀⢰⢶⠚⢳⣿⢀⣁⣾⣿⣿⣿⢉⡿⡏⣧⣿⣟⠁⠘⠀⣸⣾⡆⣿
⠀⠈⠏⠲⠤⣾⠅⣼⠎⢩⡏⠙⢿⡿⡇⠀⠀⢸⠷⢣⣿⣄⠉⡿⠎⠟⠁⣿⠃⢠⣿⣽⣯⣿⣿⣿⣯⣭⣧⣿
⠀⠀⠀⠀⢰⣿⡈⣿⠀⣿⠁⠀⠀⣆⡇⠀⠀⢠⠞⡱⠘⠋⠀⠀⠀⠀⠀⠁⠀⢡⠟⢻⡿⣿⣿⣿⣷⣿⡇⣿
⠀⠀⠈⠉⠉⠁⠉⠈⠉⠉⠁⠀⠀⠉⠉⠂⠒⠊⠒⠙⠒⠒⠒⠓⠐⠒⠂⠐⠐⠋⠐⠃⠖⠚⠟⠛⠛⠛⠛⠃

⠀⢀⣠⣤⡄
⠀⠹⠿⠿⠛⠀⠀⠀⡀⠀⠀⣤⡄⣀⢠⣀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠂⠋⠆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢀⣴⢿⣿⣿⠻⣇⠀⠀⠀⠀⢂⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠘⢀⣴⣤⣿⠯⣼⣿⡂⠀⠀⠀⠘⠀⠀⠀⠀⠈⢣⡴⢤⡀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⣀⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠟⠉⠸⠃⠤⣊⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⠀⠀⠴⠆⠈⠀⠀⠆
⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠰⠂⠁⠀⠀⠀⢒⣾⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠐⠂⡸⡇⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠑⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠖⠂⠀⠀⣀⣀⣠⠀⠀⠀⠀⠀⠈⠛⠿⠗⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠃⣀⠀⡲⢄⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⢋
⠀⠀⠀⠐⠀⠀⠀⠀⠁⠛⢠⣳⡈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠋⠉⠛⠰⠆
⠀⠀⠀⠈⢑⡀⠀⠀⠀⠷⡄⢄⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠞⠋⠀⠀⣀⠀⠀⠀⠒⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠐⠀⠁⠀⠀⠀⠘⠁⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⡈⠛⠠⣄⠀⠀⠀⠢⡍⠀⠙⠀⠀⠀⠒⠁⠀⠀⠀⢀⡀⠀⠀⠀⠘⣤
⠀⠀⠀⠘⠛⠆⠐⠂⠀⠀⠀⠀⠀⠀⡀⠉⠠⠀⠀⠀⠀⠛⠃⠠⠄⠀⠀⠀⠐⠈⠁⠀⠀⠐⠒⠂⠀⠀⠀⠀⠈⠀⠀⠀⠐⠟⢠⡄
⠀⠀⠉⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠻⣿⣿⣶⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠈⠛⠛⠛⠿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⡿⠛⠋⠉⠁⠀⠀⠈⠉⠙⠻⢿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠛⠉⠀⠈⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣏⠉⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⠀⠀⠀⠀⠸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⢀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠋⠈⠉⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏⠀⠀⠀⠀⠀⠈⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⣠⣴⣿⣿⠟⠁
⠀⢀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣷⣿⣿⡿⠛⠁
⠀⢿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠉⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠉⠉⠙⠛⠛⠉⠉⠁
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⢻⣿⣷⣤⣤⣤⣀⠀⠀⢠⣿⣿⠇
⠀⠀⠀⠀⠀⠙⠻⠿⠿⢿⣿⣷⣄⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⡤⠒⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⣰⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡇⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠲⡄
⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣈⡆⠀⠀⠀⠀⠀⠀⡸⢭⣙⣳⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⢢⡀⠀⠀⠀⠀⠀⢸⠤⠖⠤⢾
⠀⠀⠀⠀⠀⠀⠀⢻⢮⣑⡏⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠘⡇⠀⠀⠀⠀⢸⡀⣀⠀⢀⡀⢈⠆⠀⠀⠀⠀⣏⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⣷⡼⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠉⢹⠁⠀⠀⠀⠀⠀⠙⣄⣍⣡⡸⠁⠀⠀⠀⠀⠀⠈⠓⡖⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢂⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⢀⣴⡁⣹⡀⠀⠀⠀⠀⠀⢠⠞⠓⠛⢤⠗⠀⠀⠀⠀⠀⠀⣀⣟⣆⡀
⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠙⠦⡀⠀⠀⢠⠞⠓⠒⠛⠓⢛⡆⠀⠀⠀⠀⢮⣀⣀⣀⣼⠄⠀⠀⠀⠀⠀⢸⠶⠤⠤⢧
⠀⠀⠀⠀⠈⠻⣍⣑⣉⣉⣊⣩⠗⠁⠀⠀⢸⠟⠛⠛⠛⣛⣛⡇⠀⠀⠀⠀⠀⢣⣶⣶⠃⠀⠀⠀⠀⠀⠺⢍⣛⣛⣛⡩⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⡇⣃⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢄⡀⠀⠀⠀⠀⠀⢀⢔⡒⣒⠢⡀⠀⠀⠀⠀⠀⢀⡤⠒⠢⢤⠀⠀⠀⠀⠀⠀⢴⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⢠⢊⡩⣍⠲⡄⠘⡄⠀⠀⠀⠀⡇⡏⣞⠶⡱⣱⠀⠀⠀⠀⠀⣏⢀⡀⢀⠀⣱⠀⠀⠀⠀⠀⣺⣿⣽⣽⣿⡇
⠀⠀⠀⠀⠀⢤⢸⠰⣀⡧⢸⣀⠇⠀⠀⠀⠀⠱⣕⣮⣭⠷⠋⠀⠀⠀⠀⠸⡍⣿⢹⡏⡿⢸⠇⠀⠀⠀⠀⣿⣿⣯⣭⣽⡇
⠀⠀⠀⠀⠀⠈⠣⣳⣬⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⢳⣻⣸⣧⣇⡟⠀⠀⠀⠀⠀⣗⣀⣈⣉⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⡀
⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⡉⢢⠀⠀⠀⠀⠀⠀⠀⣠⡼⣝⢢⡀⠀⠀⠀⠀⠀⢺⢀⣀⡅⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⢸⠁⠏⠀⢀⠀⠈⡆⠀⠀⠀⠀⢠⠞⡄⢣⢸⠋⠁⠀⠀⠀⠀⢎⡉⠀⠈⢋⡱⠀⠀⠀⢀⡾⡀⢣⠀⣰⢃⡼⡆
⠀⠀⠀⠀⠀⠘⢄⣀⠀⢀⣁⡼⠁⠀⠀⠀⠘⣍⠢⡵⠜⠁⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠻⡀⠀⠀⠀⠀⠳⢼⠍⠉⠉⣽⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠋⠙⠦⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡗⠒⠀⠀⠀⠀⢰⡋⠉⡆⠀⣠⠞⠋⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠑⠚⠁⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⢸⠀⢠⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣤⣤
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⢸⠀⠀⡏⠉⠉⠉
⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇
⠀⢠⠇⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⣇⣀⣀⣀⣀
⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇
⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢈⠀⢸⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇
⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⡧⠤⠤⠤
⠀⠃⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠀⢸⠀⠀⠧⣄
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢀⣴⣶⣿⣷⣶⣄⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢰⣿⡟⠉⢻⣿⣿⣿⣷⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢺⣿⣿⣶⣿⣿⣿⣿⣿⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠘⢿⣿⣿⣿⣿⣿⣿⠟⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠙⠛⠿⠟⠛⠁⢀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣤⡤⠔⠚⠁
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⣀⠀⠀⠀⠀⠀⠀⠐⠂⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣾⠇⣸⡣⠀⢨⠆⠀⠀⠀⢰⣤⣴⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⢛⢻⣿⠾⢹⣏⣄⣸⣇⢤⠀⠀⠀⢙⣿⣿⣿⣧⠀⠀⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠴⢿⣿⣿⣿⣷⣾⣯⣤⠤⠤⡍⠀⠤⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⢀⡄⣶⣿⣿⣧⣨⠟⠛⠉⠏⠋⠒⠲⠒⠡⠞⣁⢾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢛⡙⠺⣫⣬⣿⣯⡀⠤⠾⣿⢷⣶⡦⠖⠒⠚⣿⣤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢿⢲⣖⣚⡯⡀⣼⠛⠁⡄⠀⠈⣎⣹⢃⣀⣠⡔⠋⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣷⢘⣿⠍⡉⢀⡰⢿⡇⢶⣿⠀⠀⡿⠛⠉⠁⠀⠱⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⠀⠀⠀⣼⣇⢹⡟⠀⠀⣇⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⣿⠞⡀⠰⠖⠄⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠠⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢦⣄⣀⣠⣄⠚⠁⡀⣊⠀⠀⡆⣀⠠⠀⠀⢠⠆⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⣄⣐⡀⢀⠰⣤⢀⣦⣾⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣟⡹⣷⣞⠛⢋⣡⣼⣯⠴⣠⣀⣴⣷⣾⠥⣤⡴⣶⣶⣋⠀⠀⠀⠀⠀⠸⡀⠉⢙⠿⢿⣼⡥⠟⡟⠀⠀⢀⡀⢴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡟⢌⠣⠈⠻⣟⠺⡿⢿⣿⡿⠃⠙⢿⣟⠛⠲⠦⠑⢄⠈⠀⠉⠀⠀⠀⠀⠀⠈⠙⠋⠀⠩⣭⣥⣤⡘⠛⢉⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⠀⠈⠐⠀⠀⠈⠻⡎⠀⠀⠹⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠶⣤⣇⠈⠙⢢⠀⠠
⠀⠀⠀⠀⠀⠀⠛⠄⠠⠀⠐⠀⠠⠄⡄⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⢰⢀⡀⠀⠀⠘⣗⠒⠒⠴⠚⠉⡩⠖⡎⢹⠻⣄⣀⢀⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠘⣧⣀⠁⢀⡜⠁⣸⠇⢠⣿⠅⠀⠈⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀⠀⠀⠀⠀⠈⢻⢀⣿⣆⣴⣿⣝⣿⣷⣶⡽⠖
⠀⠀⠀⠀⠀⠀⠀⠀⡃⢠⡾⠋⢠⠗⡄⠀⠀⠀⠀⠂⠂⠒⠆⠁⠒⠀⠀⣀⡉⠀⠀⠀⠀⠀⠀⢻⣿⠟⠉⠀⠀⠈⠱⠆⣀⡀
⠀⠀⠀⠀⠀⠀⡀⠈⡉⠹⣧⣀⣈⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠟⢧⣀⡄⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠇⢀⠀⠁⢄
⠀⠀⠀⠀⠀⠉⣀⢀⠴⠒⠬⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠶⠂⠀⠀⣀⣙⣩⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⡀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠈⠑⠉⠁⠀⠠⠀⠀⣴⣇⡀⠀⠀⠀⣠⡆⠀⡷⠀⠀⣸⣿⢛⠈⠀⠀⠀⠀⠀⠈⠀⠰⠀⠀⠀⢐⠀⠀⠹⣦⣀⠉⠄
⠀⠀⠀⠀⠀⠚⠃⠈⣬⡀⠀⠀⠁⠀⣹⠇⡀⠀⠀⠀⢻⡑⢾⣥⠄⠀⣉⠉⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠙⠓⠾⡆
⠀⠀⠀⠀⠐⠻⠛⡹⠿⢿⡀⠀⠏⠛⠁⠀⠀⠀⠀⠀⠙⣍⡀⠀⠀⠀⢿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠠⠂⡴⠉⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠆⢲⡶⠶⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠈
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠋⣿⡀⠀⠀⠀⡀⠀⠀⡔⡄⢠⡀⣠⠤⠂⠔⠂⠀⠀⠀⠀⠄⠀⡀
⠀⠀⠀⠈⠀⠠⠤⠀⠀⢠⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆⠙⠃⠀⠀⠈⢇⠀⣴⠏⡇⢸⡾⠋⠐⠀⠄⠀⠀⠀⠀⠨⠆⢠⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣀⣤⠀⠀⠀⠁⠀⠀⠀⢰⠄⠀⡆⣻⠏⢸⡇⣸⣅⠈⢀⠀⠀⠀⠀⠀⠀⠂⠀⠲⠓⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠔⠋⠙⠀⡁⠀⠀⠀⠀⠀⠀⣬⡄⠀⠀⡿⢀⣾⡧⢿⢁⣀⣤⣀⠶⠋⠁⠀⠢⠈⠀⢢⡀
⠀⠀⠀⢰⡆⢄⣀⠀⠀⠀⠀⠀⠀⢀⡆⡄⢠⠄⠀⠐⠀⠀⠀⠀⠀⠀⡀⢷⠀⠀⡇⢺⣿⡷⠛⠁⢁⡠⠜⠀⠀⠀⠐⢀⡄⡀⢀⢠⣴⡲
⠀⠀⠀⢐⠀⠢⠀⠉⠐⣄⠀⠀⣄⣸⡧⢠⡞⠀⠀⠘⠀⠀⠀⠀⠀⠀⠐⣿⡄⠁⢹⣿⢋⣠⠴⠚⠋⠀⠀⠀⠀⠐⡶⠟⠛⡠⠛⣈⠉⠶⠂
⠀⠀⠀⠀⢦⠀⠁⡆⣀⢸⡆⢸⣢⡈⣿⣿⠃⠀⠀⠃⠀⠀⠀⠀⠀⡀⠀⠘⣇⣠⣼⣿⠛⢁⠀⠀⣂⡐⠞⠀⠀⠀⣀⡞⠀⠁⠞⢁⠀⡀
⠀⠀⠀⠀⠈⠻⢶⣿⣿⣿⣿⡀⢻⡷⣿⣿⣤⣦⣤⣤⣄⡀⠀⠀⠈⢧⡁⠄⠈⢿⣏⣷⣿⡷⢞⡿⠀⠀⠀⠀⠀⠐⠈⡉⠈⡸⠀⠘⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢙⣻⣿⣿⣿⣿⣿⡟⠙⠛⠯⡉⠖⠀⠀⠳⣿⡄⠀⡸⠛⠛⠁⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⠏⣁⢌⣉
⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣉⢘⣉⢸⣻⡿⢋⣝⠶⠄⠊⠀⠀⠀⠀⠀⠘⢿⣦⡃⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠃⠐⠁⠒⠂
⠀⠀⠀⢀⣾⡿⠛⠛⠉⠉⢙⠛⠋⢹⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠉⠓⠒⠒⠒⠋⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠨⠀⠀⢠⠀⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠌⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠒⠤⠤⠤⠖⠁⠀⠀⠀⢀⣀⣀⣀⣠⡄⠀⠁⠀⠀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠉⠙⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠄
⠀⠀⠀⠀⠀⠀⢤⢤⡠⠤⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆⠀⠀⠙⠦⠶⠲⠤⣄⡀⠀⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠃⢠⠀⢀⡠⢀⡠⢲⠄⡽⠃⠀⢀⣀⣠⣤⣦
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠉⠉⠀⠀⠀⠈⠀⠁⢀⡥⠞⢁⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠳⢤⡀⠀⠀⠀⠀⠀⠀⠘⣏⠉⠀⡴⠋⠀⠈⢳⡀
⠀⠀⠀⠀⠠⢴⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠉⠳⠤⣄⣀⣠⣤⣤⣸⡆⣸⢁⠎⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⠢⣄⠀⢀⡴⠋⠉⢧⠀⠀⠀⠉⠛⠈⡴⠀⠀⠈⠳⠦⣄
⠀⠀⠀⠀⠸⠶⠦⠤⠈⡇⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⣹⠋⣷⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠐⠒⢂⣀⣀⡈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⠀⠀⠀⢀⡤⠟⠂⢀⡇⠀⠸⡄⠀⠀⠘⢦⠀⠀⠀⠀⠀⣠⣀⠀⠀⠐⣃⣠⠷
⠀⠀⠀⠀⠀⢠⡴⠶⠚⠋⠉⠉⠀⠉⢑⣾⣩⠴⠿⠶⠿⡇⠀⠀⠹⣄⠀⠀⠈⠳⣄⠀⠀⠀⣸⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠐⠒⠁⢂⡀⠀⠀⠀⠈⢦⡀⠀⠀⠈⠳⣄⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠹⢦⣀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠙⠦⣄⡀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⢀⡄⠀⠀⠀⠈⢹⡄⠀⠊⡐⠀⣸⠖⠶⠤⠴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠸⣇⠀⡜⡠⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⣆⠉⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁


⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠁⠉⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⢨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠿⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠛⠋⠉⠙⠛⠻⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣨⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠰⠿⠛⠛⠋⠉⠀⠀⠙⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢷⣄
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣴⣿⡿⠛⠛⠛⠿⠿⢿⣶⣦⣄⣀⠀⢀⣴⣿⠏⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠟⠋⠁⢀⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠁
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢿⣿⣿⡏⢀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⣹⣿⡿⠶⢶⣶⣶⣶⣶⣶⠾⠿⠛⠁
⠀⠀⠀⠘⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣴⡿⠋⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⣰⣿⠋
⠀⠀⠀⠀⢸⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⣀⣠⣴⡶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠸⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣋⣁⣀⣀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛
⠀⠀⠀⠀⠀⠹⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣟⠛⠋⠉⠁⠀⠀⢀⣀⣠⣤⣶⡿⠟⠋
⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠿⠿⢿⣿⣿⠿⠿⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣀⣤⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣤⣤⣶⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉




⣿⣿⣿⠿⣿⡿⢿⢿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣷⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡏⠛⠛⠉⠉⠉⠉⠉⢉⡉⢉⠩⠉⠉⠉⠉⠉⠉⠉⠉⠉⠃
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⡀⣀⠀⠀⠀⠀⢰⡆⢰⠀⠀⠀⠀⠀⠀⠀⣀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢄
⡇⠈⠒⠃⠚⠀⠈⠉⠉⢸⡄⢸⠀⠉⠉⠁⠀⠐⠂⠛⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠀⠁⠀⠀⠉⠒⢦⡀
⡇⠐⠂⠃⠜⠀⠁⠈⠉⢰⠆⢸⠀⠉⠉⠀⠀⠐⠀⠟⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡐⣁⡀⣠⢤⠤⡄⡄⣤⢠⣄⢙⣆
⡅⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡔⠋⠈⠉⠁⠀⢀⣀⣀⣀⡀⣀⣈⣀⡉⣣
⡇⠘⠓⠃⠛⠀⠀⠀⠀⢸⠃⢸⠀⠀⠀⠀⠀⠘⠃⠛⠀⡇⠀⠀⠀⠀⠀⠀⣥⠐⠐⠀⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⡇⠀⢀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⢠⣌⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐
⡇⠀⠈⠿⠃⠀⠀⠐⠀⢨⠀⢸⠀⠐⠒⠀⠀⠻⠌⠂⠀⡇⠀⠀⠀⠀⠀⡀⢰⠀⠈⠛⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠈⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢳⠀⠳⡄⠀⢀⣀⠠⠤⠐⠀⠀⠀⠀⢀
⡇⠀⠼⠣⠇⠀⠒⠒⠒⢈⠀⢸⠀⠒⠒⠂⠀⠿⠧⠀⠀⡇⠀⠀⠀⠀⠀⠈⠁⠀⠈⠓⢤⣀⣀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠢
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠉⠉⠁⠀⠀⠀⠀⠀⣀⣀⠤⠄⣤
⡇⠀⠀⠛⠃⠀⠒⠒⠒⠀⠀⢸⠀⠒⠒⠂⠀⠙⠛⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠀⣀⣤⣲⣄⣭⡟⠁⠉⠻⠀⠀⠁⠆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠄⢸⢀⡤⢄⠀⡇⠀⠀⢸
⡇⠀⢴⠴⠇⠀⠒⠒⠒⢈⠀⢸⠐⠒⠒⠂⠀⣤⠼⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠌⠳⣄⡼⠉⠐⢦⠇⡇⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡘⠀⠘⡇⠀⣄⠀⠀⢣⠀⣀⣴⣷⡄⠂
⡇⠀⠸⠿⠇⠀⠒⠒⠒⢀⠀⢸⠐⠒⠒⠂⠀⠷⠽⠂⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡑⠀⠈⠻⡀⠏⠀⠀⠈⠻⣿⣿⠋
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢈⠀⢱⣎⠀⠰⠀⠀⠈
⡇⠀⢤⣴⠇⠀⠒⠒⠒⠀⣶⣾⠐⠒⠒⠂⠠⣤⡄⠆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢸⠀⠀⢻⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢸⠀⠀⠀⠁
⡇⠀⢰⢴⠆⠀⠒⠒⠒⠀⣿⣿⠀⠒⠒⠂⠀⡤⠤⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠃
⡇⠀⣰⠄⠆⠀⠀⠀⠒⠀⣿⣿⠀⠂⠐⠀⢀⡆⡖⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⠰⢄⠆⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠠⠀⡶⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⠀⢀⠆⠀⠤⠤⠤⠀⣿⣿⠀⠤⠤⠀⠀⢰⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⢸⡅⠐
⡇⠀⠀⢴⠆⠀⠤⠤⠤⠀⣿⣿⠀⠤⠤⠄⠀⠀⣤⠆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⡁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⡇⠀⠀⠀⠀⠀⠀⢀⣨⠟⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⡇⠀⠀⠀⠀⠀⠀⣤⣟⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⢡⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⡇⠀⠀⠀⠀⠀⠀⠀⠙⠋⠿⠭⠉⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⣇⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⡄⠴
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣤⣤⣭⣥⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣤⣾⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀
⠀⠀⣴⣿⣿⡆⠀⠀⠀⠀⠀⠀⣤⣼⣠⠀⠀⠀⠀⣤⢸⢰⢀⠀⠀⠀⠀⠀⢠⣄⠀⢀⡤⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠰⣄⠀⢀⣾
⠀⠀⢻⣿⣿⡅⣠⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⠀⠀⠚⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⢠⣟⡁⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⢲⣽⣧⣾⣿⡴
⠀⠀⢸⣿⣿⡿⠃⠀⠀⠀⠀⡄⢺⡏⠀⠀⠀⠀⠀⠈⣿⣿⠇⠀⠀⠀⠀⣦⢨⣿⠘⠋⠀⠀⠀⠀⠀⡘⡇⠀⠀⠀⠀⠀⢀⡿⢻⡿⠟
⠀⠀⢿⣿⣿⡁⠀⠀⠀⠀⠀⢿⣾⡇⠀⠀⠀⠀⠀⠳⣿⣿⠀⠀⠀⠀⠀⠘⢿⣿⠀⠀⠀⠀⠀⠀⠀⠙⣿⣤⠀⠀⠀⠀⠈⠀⢸
⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠘⠻⡏⠁⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠿⡿⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢿

⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⢀⣐⣇⡀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣀
⠀⠀⢰⣌⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⣾⡇⠀⠀⠀⠈⣷⢸⡇⠀⣴⠀⠀⠀⠀⣿⣇⣴⡇
⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⣠⡦⡀⠀⠀⠀⢾⣿⣿⡇⠀⠀⠀⠀⠘⢿⡇⣼⡟⡀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⢸⣿⣿⣿⠂⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⣙⣷⣸⣷⣯⠶⠄⠀⠀⠀⢸⢿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⠀⠀⠀⣽⣽⠁
⠀⠀⢸⣿⣿⣿⠂⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠁⠀⢸⡏⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢸⡿⠋⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠈⠉⠛⠙⠁⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠰⣆⠀⣿⠀⣠⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠰⣄⣸⣧⣠⠀⠀⠀⠀⠀⢀⠀⣀⢀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢿⣷⣿⣾⡏⠀⠀⠀⢸⣷⡀⣰⡶⣦⠀⠀⠀⠈⠻⡿⠃⠀⠀⠀⠀⠀⢸⡄⣿⣸⠀⠀⠀⠀⠀⣪⢿⡷⠦⠀⠀⠀⠀⢀⣿⣇⡀
⠀⠀⢺⣿⣿⣿⠄⠀⠀⠀⢠⣿⣧⡟⠀⠈⠀⠀⠀⠀⠛⡇⣤⠀⠀⠀⠀⣤⣸⣿⣿⣿⡤⠀⠀⠀⠉⠁⠈⠈⡄⠀⠀⠀⠀⢸⢠⠟⡕⠄
⠀⠀⠀⠈⣍⠁⠀⠀⠀⠀⠚⠀⠻⠁⠀⠀⠀⠀⠀⠀⠛⠈⠀⠀⠀⠀⠀⠀⠙⠻⢻⡿⠇⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠃⠚
⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣠⣶⣤⣀⠀⠀⠀⠀⢸⣧⠀⢀⣤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠃⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⣿⣻⣿⡇⠀⠀⠀⠸⣿⣠⣿⠏⠀⠀⠀⠀⢠⣾⣿⣤⠀⠀⠀⠀⢀⣠⣿⣿⠀⠀⠀⠀⣠⣾⣿⣿⠏
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠹⡿⠋⠀⠀⠠⢈⣻⣿⣿⣿⣿⣷⢆⠀⠀⢸⣿⡿⠁⠀⠀⠀⢸⠟⠉⢸⣿
⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠁⠈⠁⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠈⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠉⠛⠛⠛⢻⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠿⡟⠛⠻⢶⣄⠀⠀⣀⣤⠶⠛⠁⠀⠀⠀⣀⣠⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠻⠤⠞⢁⡤⣄⠀⠙⢷⡟⠋⠁⠀⢀⣠⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⣠⣤⡀⠀⠀⢾⣠⡞⠀⢰⠞⣷⣴⠶⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⡆⠀⢧⣀⡿⠀⠀⠀⠀⠀⠀⠈⠛⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠷⠞⢁⠀⠀⠀⣴⠦⡄⠀⢴⠋⣳⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢸⡋⣹⠄⠀⠙⠦⠿⠀⠈⠉⠉⣀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⠀⠉⠁⠀⠀⠀⠀⣀⣄⠀⠀⣼⢩⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣏⣹⠂⠀⡞⢻⡆⠀⣾⣁⡼⠃⠀⣹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠋⠀⢻⣅⠀⠀⠙⠋⢁⡀⠀⠁⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉⠀⠀⣀⡴⠋⠙⢶⣄⣀⣰⣏⣹⣧⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⣤⠞⠋⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠈⢹⡏⠉⢹⡇
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠸⣧⣀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉⠉⠛⢶⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⢀⣠⣤⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⡶⠟⠉⠈⠻⣦⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠛⢛⣛⣛⣛⣛⡛⠛⢛⣛⣛⣛⣛⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣏⠀⢠⣿⣿⣿⠠⡏⠀⢠⣿⣿⣷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠀⠈⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⡔⠚⢛⣒⣳⣦⢀⡖⠛⢛⣚⣲⣦⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠳⠦⠼⠿⠿⠛⠀⠳⠤⠾⠿⠿⠋⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⣠⠤⠤⠤⢤⣄⠀⣠⠤⠤⠤⢤⣄⢸
⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣼⣤⣡⣀⣀⣛⣛⣛⠘⢇⣀⣸⣿⣿⡿⢸
⠀⠀⠀⠀⠀⢰⣿⡿⠿⢿⣿⣿⡿⠿⢿⣿⣿⡿⠿⢿⣿⣿⣿⠿⢿⣿⣇⣀⣀⣀⣀⣀⠀⢸
⠀⠀⠀⠀⠀⢸⣿⣠⠞⢁⣿⣿⣠⠔⠉⣿⣿⣁⠔⠋⣿⣿⣇⠴⠋⣿⣿⣿⣿⣭⣭⣽⣷⢸
⠀⠀⠀⠀⠀⢸⣿⣿⣶⣿⣿⣿⣿⣶⣾⣿⣿⣿⣶⣾⣿⣿⣿⣶⣾⣿⢸⣿⣿⡟⠛⠛⠁⢸
⠀⠀⠀⠀⠀⢸⣿⠟⢉⡿⣿⣿⠟⢉⡿⣿⣿⠟⢉⡻⣿⣿⡟⠋⣹⡟⠘⠛⣛⣷⣶⣖⣦⣸⣀⡀
⠀⠀⠀⠀⠀⢸⣿⣖⣉⣠⣿⣿⣶⣋⣠⣿⣿⣶⣋⣀⣿⣿⣷⣊⣠⡇⠀⡞⠉⠀⠀⢸⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠈⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢦⣷⣀⣀⣀⣈⣭⣭⣽⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀


⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠂⠀⢀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠃⠐⠀⠀⠀⣀
⠀⠀⠘⠇⠀⠀⢀⠀⠀⡀⠀⠀⠀⢀⣀⠀⠈⠀⠐⠀⠠⠐⠀⡇⠀⡀⠀⠈⠀⠁⠁⠀⠀⠐⢦⡥⠀⠀⠰⠆⠀⠀⠀⠀⠀⠠⡠⠀⠀⡀⠠⠀⠀⡀⠀⠀⠀⠀⠠
⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠄⠀⠈⠀⢨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⡑⠀⠀⠀⢃⠀⢀⠜⠁⠀⠀⠀⢆⡄
⢰⡄⠀⠀⠀⠀⠀⠁⠀⠁⠀⠄⠀⠈⠀⡄⠀⠀⠻⠀⠀⠀⠀⠈⠁⠀⠄⠀⠀⠀⠀⠀⢠⡜⡀⠀⣾⡟⠃⡀⠀⠀⣀⢀⡴⢀⠀⠀⣗⠂⠀⣌⡄⢀⣀⡔⢀⠀⣠
⠀⠀⠀⠈⠀⠀⠀⠀⠐⢀⠀⠀⠀⢠⣈⠀⠀⢀⠆⢱⡆⠀⠀⠀⠀⠘⠚⠀⠀⣀⠀⢀⣼⢱⡟⣦⡇⣻⣷⡧⠀⣲⣧⣾⢓⡤⣅⣸⢻⣦⡖⣸⣇⣿⣿⣿⣄⣸⣏
⠀⠀⡦⠀⠂⠀⠀⢠⠀⠈⠆⢀⠠⠘⠀⠁⠃⢸⡞⠂⣰⠀⠀⢸⣆⡆⣿⣶⡦⣇⣰⣿⠇⣾⣷⣿⣷⣿⣿⣧⣗⣻⣿⡟⡺⣧⣿⣿⡹⣿⣇⣿⣿⢶⣿⣿⣿⣿⢹
⠀⠀⠁⢠⡀⠧⠀⠾⠀⠴⠀⢸⡀⠀⠀⡄⠸⣞⠁⠎⢙⠰⢠⢸⡇⣧⢿⢙⡇⣿⣿⣿⢲⡿⣿⢿⡟⢺⡟⣼⣿⢻⣿⠓⣿⣿⢿⣿⠀⣿⠹⠟⣿⢸⣿⠿⠿⠣⠿
⠄⠁⠀⣼⠀⠀⡄⠀⠈⠄⠀⡾⢉⡆⠀⠀⡇⡜⢰⡷⢸⡄⢸⠘⡗⡿⠯⡞⠓⠙⠉⠋⡘⠃⠏⠀⠀⠈⠀⠉⠉⠀⣀⠐⠀⠀⠌⠀⠀⠁⠀⢠⡀⠄⠀⠀⠀⠠
⢠⠀⡏⢸⠒⠀⢃⠀⢡⣀⣸⡇⠀⣻⣀⠀⣳⠃⣬⣿⠸⡏⢹⣖⠥⠇⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠳⠀⣠⠤⠀⡀⠀⠈⠃⠀⡉⡀⠂⠀⠬⠄⠀⠐⠃⠈⠉
⢙⠢⣷⠀⡄⠀⡘⡃⠘⡁⠸⣇⢰⡇⠃⠀⠛⠠⠗⠿⠀⡇⢾⣿⣄⠀⡀⠀⠀⠀⠃⠀⠤⠀⣃⠀⣤⡀⠀⢀⡀⠀⠀⠄⠀⠀⠀⠁⠀⠀⢀⣄⣀
⡿⢸⣿⠀⡇⠀⡇⠁⠀⠈⠀⠉⠁⠀⠐⠀⢒⣂⡀⠀⠀⠀⠈⠁⠀⠀⠤⢄⡈⠁⠠⠤⠀⢀⢈⠠⠄⠤⠆⠁⢀⠀⠀⠴⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⡆⠩⡇⠂⢱⠀⠐⠆⡁⠠⠤⠀⠄⠀⠀⠀⠀⠀⠐⠂⠤⣀⠀⣀⣈⠀⠀⠀⠀⣗⡀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒
⠁⠀⠀⠁⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠉